THE CONTINUOUS-TIME PRINCIPAL-AGENT PROBLEM WITH MORAL
HAZARD AND RECURSIVE PREFERENCES.
By
Sumit Kumar Sinha

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Statistics
2011

ABSTRACT
THE CONTINUOUS-TIME PRINCIPAL-AGENT PROBLEM WITH MORAL
HAZARD AND RECURSIVE PREFERENCES.
By
Sumit Kumar Sinha
The thesis presents a solution to the principal-agent problem with moral hazard in a
continuous-time Brownian filtration with recursive preferences, and pay over the contract’s
lifetime. Recursive preferences are essentially as tractable as time-additive utility because
the agency problem induces recursivity in the principal’s utility even in the time-additive
case. Furthermore, recursive preferences allow more flexible modeling of risk aversion.
The thesis develops various results on Backward Stochastic Differential Equation (BSDE),
Functional Ito Calculus, an extension of Kuhn-Tucker Theorem and a maximum principle
for multi-dimensional BSDEs. These concepts in conjunction with the theoery of gradient
and supergradient density are used to derive a first order condition for the principal-agent
problem.
Various examples have been worked out with closed form solutions. The thesis also
presents applications of Functional Ito Calculus in Finance. Various other problems of Financial Economics such as Pareto Optimality, Altruism, direct utility for wealth are solved
using the technique developed under recursive preferences. The theory developed will be
very useful in further development of BSDE applications and Functional Ito Calculus in
financial mathematics.

DEDICATION
To my family

iii

ACKNOWLEDGMENT
I would like to express my sincere gratitude to my advisors Prof. Shlomo Levental and
Prof. Mark Schroder for the continuous support of my Ph.D study and research through
their patience, motivation and immense knowledge. Their guidance helped me through the
research and writing process of this thesis.
Besides my advisors, I would like to thank the rest of my thesis committee: Prof. Hira
Koul and Prof Mark Meerschaert for their useful comments and discussion, academically
and otherwise.
My sincere thanks also go to Prof. Hira Koul, Prof. R.V. Ramamoorthi and Prof.
Tapabrata Maiti for their help and support during my stay at Statistics and Probability
Department, MSU.
Without all of their guidance and support this thesis would not have been possible.
A very special thank you goes out to my family. To my parents, for believing in me;
without their support I would not be here. To my siblings, Vinit, Namita, Shweta, back
home and here; wherever they maybe, they always seem close to me.
Finally, a special shout out to all friends at MSU who were a great support. A special
thanks to Gaurav, Shaheen and Shalini for being patient in listening to whatever I had to say.
And then to my cookout buddies, Aritro, Avinash, Mohit, Satish and Venkat, who always
complained about my food, even though it was the best they ever had. I am also deeply
indebted to Nikita, without whom it would have been difficult to finish what I started.
Last but not the least, I would like to thank Neeraja and Chandni, two people who are
very close to me, for all their support during my stay at MSU, even when they were not
near, I could always count on them.
You all sprinkled large portions of fun into the last five years. You helped me through
this process in a country far away from home by bringing home a little closer to me.

iv

TABLE OF CONTENTS

1

2

3

4

5

Introduction . . . . . . . . . .
1.1 Thesis overview . . . . . . . . .
1.2 Principal-Agent Problem . . . .
1.3 General Maximization Problem .
1.4 Functional Itô Formula . . . . .

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Some preliminary concepts from Optimization and Financial Economics
3.1 Introduction to Optimization under constraints . . . . . . . . . . . . . .
3.1.1 Kuhn-Tucker Theorem . . . . . . . . . . . . . . . . . . . . . .
3.2 Generalized Recursive Utility . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Recursive utility and BSDEs . . . . . . . . . . . . . . . . . . .
3.2.2 Utility supergradient and gradient density calculation . . . . . .

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General Maximization Principle . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . .
4.2 General Maximization Principle . . . . . . . .
4.2.1 Optimization Problem . . . . . . . . .
4.3 Translation Invariant (TI) BSDEs . . . . . . . .
4.4 Pareto optimality under linked recursive utility .
4.5 Optimal consumption with altruism . . . . . .
4.6 Optimal portfolio with direct utility for wealth .

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Functional Itô Calculus . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . .
2.2 Notation and Definitions . . . . . . . . . . . . . .
2.3 Functional Itô Formula . . . . . . . . . . . . . . .
2.4 Applications . . . . . . . . . . . . . . . . . . . . .
2.4.1 Optimal Control . . . . . . . . . . . . . .
2.4.1.1 Example - Optimal Portfolio . .
2.4.2 Equilibrium Consumption and Risk Premia
2.5 Comparison with Dupire’s setup . . . . . . . . . .

Continuous Time Principal-Agent Problem .
5.1 Introduction . . . . . . . . . . . . . . . .
5.2 Setup and Statement of the Problem . . .
5.3 Agent Optimality . . . . . . . . . . . . .
5.4 Principal Optimality . . . . . . . . . . . .
5.4.1 Utility gradient density approach .
5.4.2 Dynamic Programming Approach
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5.5

5.6

Translation-Invariant Preferences
5.5.1 Optimality . . . . . . .
5.5.2 Quadratic Penalties . . .
Appendix . . . . . . . . . . . .
5.6.1 Proofs . . . . . . . . . .
5.6.2 Derivation of Examples

Bibliography

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100
104
109
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126

1
Introduction

1.1

Thesis overview

The thesis deals with the three problems that belong to the intersection between stochastic
analysis and financial economics. The problems are:
1. Continuous-time Principal-Agent Problem with moral hazard under recursive preferences.
2. General maximization principle with application to problems that arise in financial
economics.
3. A Functional version of Itô formula.
1

The problems will now be described in greater detail together with the statistical/mathematical
techniques that are being used in solving them.

1.2

Principal-Agent Problem

In economics, the principal-agent (owner-employee) problem deals with the difficulties
that arise when a principal hires an agent to pursue the principal’s interest. This problem
is found in most owner/employee relationships. The most common example is when the
owner (shareholders or principal) of a corporation hires an executive such as a Chief Executive Officer (CEO or agent). The term ’principal’ and ’agent’ is frequently used in the
economics literature. The specific problem that is the focus here arises when a principal
gets into a contract to compensate the agent for performing certain work that is useful to
the principal and agent’s effort is noncontractible, i.e. the agent’s effort is not part of the
contract. In the basic version of the problem, the agent chooses an effort scheme in order
to maximize his/her utility. The principal, in turn, offers a compensation contract for the
agent in order to induce the agent to perform his/her duty in a way that will maximize the
principal’s own utility. The principal moves first by offering compensation package to the
agent, who in turn selects the optimal effort scheme from his/her point of view.
The above problem has a moral hazard component, in the sense that the agent’s effort
is not part of the contract and the principal does not always observe the agent’s effort. So
as explained above the principal has to choose carefully the compensation package that
is being offered to the agent.

The goal is that the effort scheme chosen by the agent

in order to maximize his/her utility will also maximize the utility of principal. Another
complication in the selection of the compensation package arises because the agent may
have employment opportunities elsewhere, so the agent’s initial utility must exceed some
fixed amount.
In the continuous-time Brownian version, first examined in [27], the impact of effort
2

choice is typically modeled by replacing the underlying Gaussian probability measure by
an equivalent one. That is, the agent’s efforts change the probability measure, which by
Girsanov’s theorem is equivalent to a change in the drift of the driving Brownian motion.
This is a convenient way to model, for example, the impact of effort on the growth rate of
a cash flow process.
In this work, both the principal and the agent are using a type of utility known as generalized recursive utility. This class of utility is defined as the solution of Backward Stochastic Differential Equation (BSDE). The advantage of this class of utility is that it allows
to break the link between risk aversion and intertemporal substitution. Preferences are
defined recursively over current consumption and future consumptions. It is well known
that a forward-backward structure is inherent to the principal-agent problem and so the
use of recursive preferences is both flexible and natural. Previous work has considered
only additive utility, which is well known to arbitrarily link intertemporal substitution and
risk aversion (see, for example, [19]). Yet time-additivity offers essentially no advantage
in tractability because agent optimality induces recursivity to the principal’s preferences
even in the additive case. The generalized recursive utility class was introduced in [34]
with the goal of unifying the stochastic differential utility (SDU) formulation of [13], and
the multiple-prior formulation of [5]. Unlike the additive class, the recursive class allows
distinct treatment of aversion to variability in consumption across states and across time.
Furthermore, the class can accommodate source-dependent (domestic versus foreign, for
example) risk aversion, differences in agent and principal beliefs, as well as first-order
risk aversion (which imposes a higher penalty for small levels of risk) in addition to the
standard second-order risk aversion.1 Also, [47] shows that SDU, a special case of the
recursive class, includes the robust control formulations of [2], [24], and [37].
Some of the statistical/mathematical techniques used to solve this problem are also used
1 See [44] and [48]. We consider only second-order risk aversion in this paper, but
extensions to the first-order case (modeled by kinks in the aggregator) can be handled
along the lines of [44].
3

in the next problem, which will be discussed briefly in the next section.

1.3

General Maximization Problem

Here, the topic of interest is the development of general maximization principle under
recursive preferences. The maximization principle is developed for the optimization of
a linear combination of recursive utilities for several agents. The theory can be applied
to solve various economic problems such as Pareto optimality, Altruism, etc.

One of

the applications in this thesis of the maximum principle is in solving the principal-agent
problem described in Chapter 4.

To introduce the maximization problem an overview

of Translation-Invariant (TI) preferences is presented. After developing the thoery, it is
applied in detail to the TI case.
The study of generalized recursive utility is based on the theory of BSDE. The theory
of stochastic calculus is the basis for the theory of BSDE. More specifically, the method
of proving existence and uniqueness of solutions of BSDE is a combination of the martingale representation theorem and fixed-point theorem. Other techniques that are being used
in the maximization problem are utility gradient and dynamic programming theory. This
technique is an extremely useful method as it provides a way of determining first order conditions for optimality. The main result is based on an extension of Kuhn-Tucker theorem,
a Lagrange multiplier type of result, that is being used here for an optimization problem in
infinite dimesion setup with restrictions formulated as inequalities.

1.4

Functional Itô Formula

The Functional Itô formula is an extension of the classical Itô formula for functionals that
are defined on the history of the process rather than the current value of the process. This
result was developed in [16] for the continuous case and was extended in [6] to the general
4

semimartingale case. The thesis develops a new and much simpler proof for the general
case. The proof of the theorem uses an original setup to the problem, which makes it easy
to link functional Itô formula with the classical Itô formula. Some applications to optimal
control and portfolio theory are presented.

5

2
Functional Itô Calculus

2.1

Introduction

Itô’s stochastic calculus has been used to analyze random phenomenon and has lead to new
fields in stochastic processes. There are many applications of Itô’s stochastic calculus in
various fields of science such as probability, statistics, mathematics etc. The traditional
Itô formula applies to functions of the current value of a semimartingale. But in many
applications, such as mathematical finance and statistics, it is natural to consider functionals
of the entire path of a semimartingale. The Itô formula was extended recently by [16] to
the case of functionals of paths of continuous semimartingales.
6

This work was motivated by the talk of Bruno Dupire at 6th World Congress of the
Bachelier Finance Society (Toronto, June, 2010) which described several applications of
the functional Itô formula.

Among the applications was an explicit expression for the

integrand in the martingale representation theorem, and a formula for the running maximum
of a continuous semimartingale.
The setting in [16] uses somewhat exotic concepts such as vector bundles and some new
types of derivatives. We modify Dupire’s setup to a standard one: The vector bundle is replaced by the usual right continuous left limit space of functions, and Dupire’s derivatives
are replaced by standard directional derivatives in infinite-dimension spaces. With this
setup the functional Itô formula comes as a simple and natural extension of the standard Itô
formula. Furthermore, this work provide a simple proof of the extension to the more general case of cadlag semimartingales (which allows for jumps in the stochastic processes).
The proof for the cadlag case in [6] uses the analytic approach developed by [20]. However
the proof here is much shorter and uses traditional and basic concepts. A statement of the
result in the special case of (continuous) Brownian semimartingales appears in [7].

2.2

Notation and Definitions

T g ; F; P ). Let Xt (!) 2 Rd ; 0
T , denote an adapted RCLL semimartingale. Let D [0; T ]; Rd be the space of RCLL

Consider the usual stochastic base ( ; fFt ; 0
t

t

Rd -valued functions on [0; T ] equipped with the supremum metric. Let F : D [0; T ]; Rd !
R be a functional. Define

Yt (s) = fX(s ^ t); 0

s

Tg;

that is, Yt represents the history of the process, X up to time t: Observe that (Yt )0 t T
is adapted to the fFt g filtration, even though Yt is defined on the interval [0; T ]:
7

The following notation will be used throughout:
8
>
< 0 if 0
1[t;T ] (s) =
> 1 if t
:

s < t;
s

T:

Also, ei i=1;::d will denote the canonical basis in Rd (that is, ei is a length-d vector
with a one in the ith position and zeros elsewhere). Then the following definition of the
directional derivatives of the functional F will be used.
Definition 2.1. For x 2 D [0; T ]; Rd

1
we denote by Di F (x; [t; T ]) the (directional)
derivative of F at x in the direction of the Rd -valued process 1[t;T ] ei . Namely,
1
Di F (x; [t; T ]) = lim
h!0

F (x + h1[t;T ] ei )

F (x)
:

h

Similarly we denote the second-order derivative in the direction 1[t;T ] ei and 1[t;T ] ej by
D1 F (x + h1[t;T ] ei ; [t; T ])
2 F (x; [t; T ]) = lim j
Dij
h
h!0

1
Dj F (x; [t; T ])

Note that time can be taken as one of the components in D [0; T ]; Rd . In general
the functional itself is not necessarily dependent on time (see Section 2:5 below). However, the derivatives defined above are functions of (x; t) 2 D [0; T ]; Rd

consider continuity of the derivatives we define a distance on D [0; T ]; Rd
e
d [(x; t); (y; s)] = kx

yk1 + jt

sj ; where kx

yk1 =

sup kx(t)
0 t T

e
fine (xn ; tn ) ! (x; t) if d [(xn ; tn ); (x; t)] ! 0:
n!1
e
d
Let
1
D1 F (x; [t; T ]) = Di F (x; [t; T ]); i = 1; :::d ;
2
D2 F (x; [t; T ]) = Dij F (x; [t; T ]); i = 1; :::d; j = 1::::d :
8

[0; T ]. To
[0; T ] by
y(t)k : De-

2.3

Functional Itô Formula

Assume throughout the following continuity condition on the functional.
Condition 2.1. The functional F : D [0; T ]; Rd ! R is continuous and Dk F (x; [t; T ]),
e
k = 1; 2, exist and are continuous in x and t: All continuity relates to the metric d: (In

e
other words, if d [(xn ; tn ); (x; t)] ! 0 then F (xn ) ! F (x) and Dk F (xn ; [tn ; T ]) !

Dk F (x; [t; T ]); k = 1; 2.)

Next, the proof for the functional Itô formula for the case of continuous semimartingales
is given. Formally this follows from the RCLL case which will be proved later, but the proof
of the continuous case shows clearly some of the ideas being using.
Theorem 2.1 (Functional Itô formula for continuous semimartingales). Let Xt be a ddimensional continuous semimartingale and F : D [0; T ]; Rd ! R a functional that
satisfies Condition (2:1). Then
d t
XZ

Zt
h
i
1 X
1
i
2
i j
Di F (Ys ; [s; T ])dXs +
F (Yt ) = F (Y0 ) +
Dij F (Ys ; [s; T ])d Xs ; Xs :
2
i=1 0
i;j d 0
(2.1)
n
Proof. Consider partitions tn ; 0
i

i

n; 1

n

o
1 of [0; T ] such that mesh of the

partition goes to 0 as n ! 1, and define a sequence of approximations for Yt as
Appn (Yt ) =

n 1
X
Yt (tn )1[tn ;tn ) ;
k
k k+1
k=0

for each t 2 [0; T ]:

For t restricted to [tn 1 ; tn ] we have
k
k
Appn (Yt ) = Appn (Ytn ) + Xt
k 1

Xtn
1h n i ;
t ;T
k 1
k

t 2 [tn 1 ; tn ]:
k
k

If t 2 [tn 1 ; tn ], then F (Appn (Yt )) = f (Xt ; A) for some function f : Rd
k
k
9

(2.2)

! R,

where the random variable A 2 Ftn . It is easy to confirm
k 1
rx f (Xt ) = D1 F (Appn (Yt ); [tn ; T ]);
k
rxx f (Xt ) = D2 F (Appn (Yt ); [tn ; T ]);
k

if t 2 [tn 1 ; tn ]:
k
k

The classical Itô formula for functions implies that (2:1) holds in [tn 1 ; tn ]; hence for
k
k
n : tn tg, the formula becomes
t 2 [0; T ]: Defining tn (t) = minftk k
F (Appn (Yt )) = F (Appn (Y0 )) +

d t
XZ

1
i
Di F (Appn (Ys ); [tn (s); T ])dXs

(2.3)

i=1 0

Zt
h
i
1 X
2
i j
Dij F (Appn (Ys ); [tn (s); T ])d Xs ; Xs :
+
2
i;j d 0
The goal is now to complete the proof of the theorem by letting n ! 1 in (2:3), relying
on the dominated convergence theorem for stochastic integrals, which is quoted for the
reader’s convenience.
Theorem 2.2 (Dominated Convergence Theorem for Stochastic Integrals). Let X be a
semimartingale. If (K n ) is a sequence of predictable processes converging to zero pointwise a.s. and if there exists a locally bounded predictable process K such that jK n j
Z
K; for all n; then K n (s)dXs ! O(p).
0

Proof. See Theorem 4.31, Chapter IV in [29].
Remark 2.1. If X is a continuous semimartingale, then K n and K can be taken as progressively measurable in the above theorem.

The following following lemma will be used for the proof of theorem.

10

Lemma 2.1. Let tn : [0; T ] ! [0; T ] be a sequence of functions with tn
I; where
I(t) = t. Let F : D [0; T ]; Rd ! R: Assume kxn xk1 + ktn Ik1 ! 0;
n!1
d . Then, under the continuity assumptions of Condition 2.1,
where xn ; x 2 D [0; T ]; R
sup
0 t T

n

Di F (xn ; [tn (t); T ])

Di F (x; [t; T ])

o

! 0;
n!1

i = 1; 2:

Proof. If not then 9" > 0; a sequence of integers nk " 1, and a sequence sk 2 [0; T ] such
that
Di F (xnk ; [tnk (sk ); T ])

Di F (x; [sk ; T ])

":

(2.4)

By moving to a subsequence we can assume without loss of generality that sk ! s ; for
2 [0; T ]: Because tn converges uniformly to I, we also get tnk (sk ) !
k!1
i F (x ; [t (s ); T ]) ! Di F (x; [s ; T ])
s . Our continuity assumption implies D
nk nk k
k!1
i F (x; [s ; T ]) ! Di F (x; [s ; T ]): This contradicts (2.4).
as well as D
k
k!1
some s

Next, the proof of the functional Itô formula is continued. Because Yt has a continuous
path it follows that (Appn (Yt ); tn (t)) ! (Yt ; t), and by our continuity assumptions, we
e
d
obtain, for each t 2 [0; T ] :
F (Appn (Yt ))
Dk F (Appn (Yt ); [tn (t); T ])

! F (Yt )
n!1

and

! Dk F (Yt ; [t; T ]);
n!1

k = 1; 2:

Observe that for each x 2 D [0; T ]; Rd , it follows from our assumptions that Di F (x; [t; T ])
is continuous in t, and hence it is bounded on [0; T ]: Therefore,

sup
0 t T

n

Di F (Yt ; [t; T ])

o

< 1; a:s:;

i = 1; 2:

Also, Di F (Appn (Yt ); [tn (t); T ]) is bounded on [0; T ] a.s. for each n: By Lemma 2.1
11

conclude that

sup
n 1;0 t T

n

Di F (Appn (Yt ); [tn (t); T ])

o

< 1; a:s:;

i = 1; 2:

Use Theorem 2.2 for each of the two stochastic integrals in (2.3). For the first and second
integrals, respectively, take
K n;1 (t)
K n;2 (t)

o
Xn
1
1
Di F (Appn (Yt ); [tn (t); T ]) Di F (Yt ; [t; T ]) ;
i d
o
X n
2 F (Appn (Y ); [t (t); T ]) D2 F (Y ; [t; T ]) ;
Dij
t n
t
ij
i;j d

with the locally bounded processes K (1) (t) and K (2) (t), respectively, where
K (i) (t) =

sup
n 1;0 s t

n

Di F (Appn (Ys ); [tn (s); T ]) + Di F (Ys ; [s; T ]

o

;

i = 1; 2:

Remark 2.2. By adding two more steps to the above proof, one can produce a proof without
relying on the classical Itô formula (obviously the classical Itô formula is a corollary of the
theorem by taking F (Yt ) = f (X(t))). The two steps are:
1. Showing that if (2.1) holds for F and G; then (2.1) holds for F G (integration by
parts formula).
2. Showing (2.1) in the case where F (Yt ) = f (Yt (t1 ); :::; Yt (tn )) by using polynomial
approximation (Weirstrass theorem) and stopping times.
Because those steps appear in the proof of the classical Itô formula, we can conclude
that the Functional Itô formula is a natural extension of Itô formula with one more step.

The next step is to prove the functional Itô formula for RCLL semimartingales.

12

Theorem 2.3 (Functional Itô Formula for RCLL Semimartingales). Let Xt be a d-dimensional
RCLL semimartingale and F : D [0; T ]; Rd ! R , a functional that satisfies Condition
(2:1). Then
d t
XZ

d Zt
1 X
i;c j;c
1
i
2
F (Yt ) = F (Y0 ) +
Di F (Ys ; [s; T ])dXs +
Dij F (Ys ; [s; T ])d[Xs ; Xs ]
2
i=1 0
i;j=1 0

+

X

s t

2

4F (Ys )

F (Ys )

(2.5)

3

d
X

1
i
Di F (Ys ; [s; T ]) Xs 5 ;
i=1

c
where Xt is the continuous part of the semimartingale Xt ;

Xt = Xt

Xt , and

8
>
< X(u) if
u < t;
Yt (u) =
> X(t ) if t u T:
:
n
o
n = k T; k = 1; :::::; n, and again let t (t) = inf tn : tn > t . For a
Proof. Define tk
n
n
k k

fixed n, define a sequence of stopping times
n
0 = 0;

n
n
k+1 = inf t > k : X(t)

1
n
X( k ) >
n

n
^ tn ( k ) ^ T:

We remark that the above sequence of stopping times can be viewed as a union of the fixed
n o
n o
n
n
n
stopping
times
tk
and
where
=
0;
0
k ;
n
o
1
n
n
n
k+1 = inf t > k : X(t) X( k ) > n ^ T . Since X(t) is a RCLL semimartinn
o
n
n , a.s. on the event
n < T : Also there exists a k < 1 such
gale, we have k+1 > k
k
n = T: Next, define an approximation of Y as follows. For t 2 [ n ; n ),
that k
t
k k+1
2
3
k 1
X
n
n
Appn (Yt ) = 4
X( l ) 1[ n ; n ) 5+X( k ) 1[ n ;t (t)) +X(t) 1[t (t);T ] (2.6)
n
l l+1
k n
l=0
13

(The bracketed term on the right side of (2:6) is 0 if k = 0.) Observe that Appn (Yt ) as
n n
defined in Equation (2:6) is adapted. If t 2 [ k ; k+1 ) then F (Appn (Yt )) = f (X(t); A)
for some function f : Rd
! R, where the random variable A 2 F n : It is easy to
k
confirm that

rx f X t

= D1 F (Appn (Yt ); [tn (t); T ]);

rxx f Xt

= D2 F (Appn (Yt ); [tn (t); T ]);

n n
if t 2 [ k ; k+1 ):

Apply the classical Itô formula for RCLL semimartingales to get (2.5) for F (Appn (Yt ))
n n
on the interval [ k ; k+1 ); and hence for [0; T ]: The formula becomes

F (Appn (Yt )) = F (Appn (Y0 )) +

d t
XZ

1
i
Di F (Appn (Ys ); [tn (s); T ])dXs +

(2.7)

i=1 0

d Zt
1 X
i;c j;c
2
Dij F (Appn (Ys ); [tn (s); T ])d[Xs ; Xs ]+
2
i;j=1 0
3
2
n (Y )) F (Appn (Y ))
F (App
s
s
7
X 6
7
6
d
7:
6 X
1
i
4
Di F (Appn (Ys ); [tn (s); T ]) Xs 5
0 s<t
i=1
The goal is now to complete the proof of the theorem by letting n ! 1 in (2.7) relying on

Theorem 2.2. First we show kAppn (Yt )(u)

Yt (u)k1

! 0: Indeed, we estimate,
n!1

for u 2 [0; T ],

Appn (Yt )(u)

8
> X( n )
>
>
k
>
>
>
>
< X( n )
k
Yt (u) =
>
> X( n )
>
>
k
>
>
>
:
jX(t)

X(u) < 2=n; if
X(u) < 2=n; if
X(t) < 2=n; if
X(t)j = 0;

14

if

n
k
n
k
n
k

n
u < k+1 t;
n
u t < k+1 ;
n
t < u < k+1 ;
tn (t)

u

T:

It follows that (Appn (Yt ); tn (t)) ! (Yt ; t) and (Appn (Yt ); tn (t)) ! (Yt ; t):
e
e
d
d
By Condition (2.1), for each t 2 [0; T ];
F (Appn (Yt ))

! F (Yt ); F (Appn (Yt )) ! F (Yt );
n!1
n!1
Dk F (Appn (Yt ); [tn (t); T ]) ! Dk F (Yt ; [t; T ]); k = 1; 2:
n!1
Next, we establish the local boundedness needed in Theorem 2.2. Exactly as in the proof
of Theorem 2.1, Condition (2.1) implies
n

sup
0 t T

Di F (Yt ; [t; T ])

o

< 1; a:s:;

i = 1; 2;

and by using Lemma 2.1 we can conclude that

sup
n 1;0 t T

n

Di F (Appn (Yt ); [tn (t); T ])

o

< 1; a:s:;

i = 1; 2:

We use Theorem 2.2 for each of the two stochastic integrals in (2.7). For the first and
second integrals, respectively, take
K n;1 (t)
K n;2 (t)

X

1
Di F (Appn (Yt ); [tn (t); T ])

i d
X

2
Dij F (Appn (Yt ); [tn (t); T ])

1
Di F (Yt ; [t; T ]) ;
2
Dij F (Yt ; [t; T ]) ;

i;j d
with the locally bounded processes K (1) (t) and K (2) (t), respectively, where
K (i) (t) =

sup
n 1;0 s t

n

Di F (Appn (Y

i
s ); [tn (s); T ]) + D F (Ys ; [s; T ]

i = 1; 2:
15

o

;

So we can conclude the following convergence in probability:
d t
XZ

i
1
Di F (Appn (Ys ); [tn (s); T ])dXs

i=1 0

d Zt
1 X
i;c j;c
2
+
Dij F (Appn (Ys ); [tn (s); T ])d[Xs ; Xs ]
2
i;j=1 0
d t
d t
XZ
X Z
i;c j;c
1 F (Y ; [s; T ])dX i + 1
2
!
Di
Dij F (Ys ; [s; T ])d[Xs ; Xs ]:
s
s 2
n!1
i=1 0
i;j=1 0
The last step is to deal with the sum involving jumps in formula (2.7). Let

gn (s) = F (Appn (Ys ))

g(s) = F (Ys )

F (Appn (Y

F (Ys )

s ))

d
X

d
X

1
i
Di F (Appn (Ys ); [tn (s); T ]) Xs ;
i=1

1
i
Di F (Ys ; [s; T ]) Xs ;
i=1

s 2 [0; T ]:

X
gn (s) !
Observe that gn (s) ! g(s); s 2 [0; T ]: So to prove that
n!1
n!1
0
X s<T
X
g(s); a.s., we need to find G(s)
sup jgn (s)j so that
G(s) < 1, a.s.
n
0 s<T
0 s<T
n (Y )) we get that 9 r (s) 2 [0; 1] such that
By the Taylor’s theorem applied to F (App
s
n
jgn (s)j

d
X

2
i
Dij F (Appn (Ys ) + rn (s) 1[t (s);T ]
Xs ; [tn (s); T ])
Xs
n
i;j=1
0
1
8
>
n (Y )
>
App
>
s
C
> D2 F B
>
@
A
>
>
ij
<
+r 1[t (s);T ]
Xs ; [tn (s); T ]
n
sup
>
d
>
X
>
j
>
i
n 1; 0 s T
>
Xs
Xs
>
>
:
i;j=1
r 2 [0; 1]; 1 i; j d
G(s):

16

j
Xs
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;

Because
X

0 s<T

X

d
X

j
Xs

i
Xs

d

0 s<T i;j=1
G(s) < 1 if we can show that

H

1; 0

r 2 [0; 1]; 1

[X i ; X i ]T < 1, a.s., we will get that
i=1

2
Dij F (Appn (Ys )

sup
n

d
X

s

+r 1[t (s);T ]
n

T

i; j

Xs ; [tn (s); T ])

< 1; a.s.

d

If not then there exist subsequences nk ! 1; [0; T ] 3 sk " s (or sk # s or sk =
s ); [0; 1] 3 rk ! r ; 1

i; j

d so that

2
Dij F (Appnk (Ys ) + rk 1[t (s );T ]
nk k
k

Xs ; [tn (sk ); T ]) ! 1:
k
k

(2.8)

However,
2
Xsk ; [tnk (sk ); T ])
Dij F (Appnk (Ys ) + rk 1[t (s );T ]
nk k
k
8
>
2
>
if sk " s ;
Dij F (Ys ; [s ; T ])
>
>
<
2
!
Dij F (Ys ; [s ; T ])
if sk # s ;
>
>
>
>
: D2 F (Y
+ r 1[s ;T ]
Xs ; [s ; T ]) if sk = s :
s
ij

(2.9)

Because Condition 2.1 implies that D2 F (x; [t; T ]) is bounded on [0; T ] for each x 2
D [0; T ]; Rd ; we conclude that (2.9) contradicts (2.8). Therefore H is finite a.s.

17

2.4

2.4.1

Applications

Optimal Control

Let (E; B (E)) be a mark space, where E is either Euclidean space, with B (E) denoting
the Borel -algebra, or a discrete space, with B (E) denoting the set of all subsets of E:
Let uncertainty be driven by a one-dimensional Brownian motion B and an independent sequence f(Tn ; Jn )g of random jump times and E-valued random jump values, respectively,
such that Tn+1 > Tn ; a.s. and limn!1 Tn = 1; a.s. Associated with the sequence
f(Tn ; Jn )g is the counting random measure p :

B ([0; T ])

B (E) ! f1; 2; : : : g,

defined as

p ([0; t] ; S) =

1
X

n=1

1 fTn

t; Jn 2 Sg ;

t

T;

S 2 B (E) ;

which denotes the number of jumps by time t whose values fall within the (Borel measurable) set S. The random measure p is known as an integer-valued point process.
The compensator of p (!; dt

dz) is assumed to be of the form h (!; t; dz) dt, for an

intensity kernel h. The corresponding compensated random measure is defined as

p (!; dt
b

dz) = p (!; dt

We can interpret h (!; t; S) as the time t

dz)

h (!; t; dz) dt:

conditional per-unit-time probability of a jump

whose magnitude falls in set S.
Let A

P (Rn ) (that is, the set of Rn -valued predictable processes) denote the set

of admissible controls.

For any control

2 A; consider the R-valued state variable

semimartingale process Xt satisfying (recall that Yt
18

represents the history of X just

before time t)

dXt =

with ;

t; Yt ; t dt +

:

[0; T ]

We assume that ; ;

t; Yt ; t dBt +

D [0; T ]

R ! R and

:

Z

b
t; Yt ; t ; z p (dt

E

[0; T ]

D [0; T ]

dz) ;
(2.10)
E ! R.

R

satisfy appropriate conditions that guarantee unique solution for

equation (2.10). (See Theorem 7 in [39] for a precise formulation of the conditions.)

We assume E

RT
0 f (s; Ys; s ) ds + g(YT )

< 1; for all

2 A and define for

any time-t starting history Yt the value function

V (t; Yt ) = sup Et
2A

Z T
t

where the dynamics of X for s
script

f (s; Ys ; s ) ds + g YT

!

;

t 2 [0; T ] ;

(2.11)

t are as in (2:10), though we henceforth omit the super-

from Y .

The agent’s problem is to choose the control process,

that maximizes the time-0

value function:

V (0; Y0 ) = sup V (0; Y0 ) = sup E
2A
2A

Z T
0

f (s; Ys; s ) ds + g YT

!

:

The following informal argument motivates the Bellman equation (2:15) below. Given
any history Yt of the process Xt , the Bellman principle in our setting is

V (t; Yt ) = sup Et
2A
where

Z t+ t

t satisfies t +

T.

t

t

f (s; Ys; s ) ds + V (t +

!

t; Yt+ t ) ;

(2.12)

Furthermore, functional Itô’s formula (Theorem 2.3)
19

implies (we assume that V satisfies Condition 2.1)

V (t +

t; Yt+ t )

V (t; Yt ) =

Z t+ t
t

D s V (s; Ys )ds +

Z t+ t
t

dMs s ;

(2.13)

where the drift operator D t and local martingale Mt t are defined by1
D t V (t; Yt ) = Dt V (t; Yt ) + D1 V (t; Yt ; [t; T ]) (t; Yt ; t ) +
1 2
D V (t; Yt ; [t; T ]) 2 (t; Yt ; t ) +
2
8
9
>
Z > V (t; Y + 1
<
=
t
[t;T ] t; Yt ; t ; z ) V (t; Yt ) h (t; dz) ;
>
E>
:
;
D1 V (t; Yt ; [t; T ]) t; Yt ; t ; z

and

dMt t = D1 V (t; Yt ; [t; T ]) t; Yt ; t dBt +
Z n
o
V (t; Yt + 1[t;T ] t; Yt ; t ; z ) V (t; Yt ) p (dt
^
E

dz) :

We assume that for all
2 A, the local martingale Mt t is a martingale (namely, we
n
o
assume M
: 2 [0; T ]; stopping time is uniformly integrable). Then we get from
(2:13)

Et V (t +

t; Yt+ t )

V (t; Yt ) = Et

Z t+ t
t

!

(2.14)

t and letting

t # 0 that,

Das V (s; Ys )ds :

Comparing equation (2:12) and (2:14) we get by dividing by

1 In
what
follows,
we
use
the
notation
Dt V (t; Yt )
1 V (t; Y ; [t; T ]); Di V (t; Y ; [t; T ]) = Di V (t; Y ; [t; T ]) for i = 1; 2.
Dt
t
t
t
Y
20

=

for any history Yt ,
0

f (t; Yt; t ) + D t V (t; Yt );

t ) + D t V (t; Yt; );

0 = f (t; Yt;
where

t 2 [0; T ] ;

all

2A

t 2 [0; T ] ,

R t+ t
f (s; Ys; s ) ds + V (t +
t

2 A satisfies V (t; Yt ) = Et

(2.15)

t; Yt+ t ) .

The only change in the following standard verification Lemma is in the drift term D t .

D [0; T ] ! R satisfy the terminal condition

Lemma 2.2 (Verification). Let V : [0; T ]

V (T; y) = g (y) for all y 2 D [0; T ]. Assume Condition 2.1 holds for V and there exists a
control

2 A for which (2.15) holds. Then

Proof. Consider any admissible process

is optimal.

2 A. By (2:15) we have

D t V (t; Yt ) =

f (t; Yt; t )

for some nonnegative process p (which is zero when

g YT

Z T

=

pt
). Therefore
Z T

dMs s
0
0
Z T
Z T
=
f (s; Ys ; s ) + ps ds +
dMs s :
0
0

V (0; Y0 ) =

D s V (s; Ys )ds +

Taking expectations we have

V (0; Y0 ) = E
E

for all

Z T

0
Z T
0

f (s; Ys , s )ds + g YT
f (s; Ys , s )ds + g YT

2 A.
21

!

!

2.4.1.1

Example - Optimal Portfolio

Let B denote d-dimensional Brownian motion, and X a wealth process for an agent trading
in n

d financial securities.

The control process

2 P (Rn ) represents the dollar

investments in the risky assets. The budget equation is
dXt = Xt r + 0 R dt + 0
t
t

R
t dBt +

Z

E

R (t; z) p (dt
^

dz) ;

X0 = x;

where x 2 R is initial wealth, r 2 R+ is the riskless rate, R 2 Rn the excess (above
the riskless rate) instantaneous expected returns of the risky assets, R 2 Rn d the return

volatility corresponding to Brownian noise, and R (t; z) 2 Rn the sensitivity of returns

to a jump of size z; z 2 E. We assume R is full rank and
Z t
0

0 R + 0 R0 R +
s
s
s

Z

0 R (s; z) h(s; dz) ds < 1
s

E

2

To rule out doubling-type strategies we assume that E 4 sup Xt
t2[0;T ]
Xt = max f0; Xt g.

a.s.

2

for all t < T:

3

5 < 1, where

The agent is assumed to maximize the expected value of some functional u : D [0; T ] !
R of the wealth process history:

max Eu YT :
2P(Rn )
We assume u satisfies Condition 2.1 as well as
(i) (monotonicity) D1 u(y; [t; T ]) > 0 for all y 2 D ([0; T ]) and t 2 [0; T ].
(ii) (concavity)

u( y1 + (1

)y2 )

u(y1 ) + (1

)u(y2 );
22

for all y1 ; y2 2 D ([0; T ]) ;

2 [0; 1]:

The time-t value function given history Yt is V (t; Yt ) = max 2P(Rn ) Et u YT .
Lemma 2.3. Assumption (i) implies D1 V (t; Yt ; [t; T ]) > 0;

t 2 [0; T ] : Assumption

(ii) implies V (t; Yt ) is concave in the direction 1[t;T ] for all t 2 [0; T ]. That is,
V (t; Yt + f h + (1

)kg 1[t;T ] )

V (t; Yt + h1[t;T ] ) + (1

)V (t; Yt + k1[t;T ] );

for all h; k 2 R,

2 [0; 1]:

Proof. To prove the first result, observe that if z(0) = 0 and z(t) is smooth and increasing,
dzt
i.e.
0; then u(y + z)
u(y); y 2 D ([0; T ]) : Indeed, by using the Functional
dt
T
R
dz
Itô formula we get u(y + z) u(y) = D1 u(Yt + Zt ; [t; T ]) t dt
0, where
dt
0
Zt (s) = z(s ^ t), and Yt (s) = y (s) if s < t and Yt (s) = y (t ) if s
t. Let
h;
YT denote the time-T wealth history from investing the additional h dollars at t in the
h;
money-market account, and note that r > 0 implies YT (s) = YT (s) + h er(s t)
h;
1[t;T ] (s) YT (s)+h 1[t;T ] (s). So we conclude that u YT
u YT + h 1[t;T ]
h;
and consequently V (t; Yth )
Et (u(YT ))
Et (u YT + h 1[t;T ] ) where the first
h;
inequality follows because YT is generally a suboptimal policy. We calculate
D1 V (t; Yt ; [t; T ]) = lim
h#0

V (t; Yth )

V (t; Yt )

h
n
Et u YT + h 1[t;T ]
lim
h
h#0
0
u YT + h 1[t;T ]
B
Et @ lim
h
h#0

= Et D1 u YT ; [t; T ]

> 0;

u YT

o

1

u(YT ) C
A

where the second inequality follows from Fatou’s lemma. For the proof of the second
part, let h and k denote the optimal portfolio processes (after time t) for V (t; Yth ) and
23

V (t; Ytk ) for any h; k 2 R. Letting
V (Yth ) + (1

2 [0; 1] then

)V (Ytk ) = Et

h;
u YT h

Et u

+ (1

h;
YT h + (1

V (t; Yth + (1

k;
)u YT h
k;
)YT h

)Ytk );

where the first inequality follows from the concavity of u, and the last follows because the
k;
h;
) k)
)YT h is feasible (with the portfolio h + (1
wealth history YT h + (1
but not necessarily optimal for Yt + f h + (1

)kg 1[t;T ] .

The concavity of V in the direction 1[t;T ] gives D2 V (t; Yt ; [t; T ])

0; but we will

assume the inequality is strict: D2 V (t; Yt ; [t; T ]) < 0. Lemma 2.2 implies that if
satisfies
0 = max Dt V (t; Yt ) + D1 V (t; Yt ; [t; T ]) Xt rt + 0 R +
t t
t
1 2
D V (t; Yt ; [t; T ]) 0 R R0 t +
t
2
Z n
o
V (t; Yt + 1[t;T ] 0 R (t; z)) V (t; Yt ) h (t; dz) ; t 2 [0; T ] ;
t
E
then

is optimal. The first-order condition
0 = D1 V (t; Yt ; [t; T ]) R + D2 V (t; Yt ; [t; T ]) 0 R R0 +
t
t
Z
R (t; z)
D1 V (t; Yt + 1[t;T ] 0 R (t; z))h (t; dz)
t
E
24

implies that the optimal portfolio t =

t = R (t; Yt )

R R0

(t; Yt ) solves the implicit equation

0

R+

1

C
1B
B
C;
1 V (t;Y +1
0 R (t;z))
A
@ R
R D
t
[t;T ] t
(t; z) E
h (t; dz)
D1 V (t;Yt ;[t;T ])
D1 V (t; Yt ; [t; T ])
where R (t; Yt ) =
:
D2 V t; Yt ; [t; T ]

In the absence of jumps ( R = 0), the agent invests in a mean-variance efficient portfolio
as in the standard model in which u is a concave function of terminal wealth only, but in our
case the scale factor R (t; Yt ) depends both on time and the history of the wealth process.
The presence of jumps distorts the investments. Supposing for simplicity that R = I and
R
jumps are Poisson distributed with intensity (that is, E = f1g and E h (t; dz) = ) then
(recall D1 V is positive)
0

(t; Yt ) = R (t; Yt ) @ R + R (t; 1)

1
+ 1[t;T ] 0 R (t; 1))
t
A:
1 V (t; Y ; [t; T ])
D
t

D1 V (t; Yt

The impact on the optimal investment of a positive potential return jump for stock i,
R
i (t; 1) > 0, is to increase the optimal investment in that stock relative to stock j with no
R
R
jump component ( j (t; 1) = 0). Conversely, the impact of a negative jump, i (t; z) < 0,
is the opposite. The effect of the jump increases with the intensity of the jumps.

2.4.2

Equilibrium Consumption and Risk Premia

A common problem in finance and economics is to derive the interest rate and risk premium process under which a given consumption process is optimal for an agent trading
in financial markets. For example, in a single-agent economy with a given endowment
process, the interest rate and risk premium processes such that the endowment is the optimum consumption process corresponds to a pure-trade asset market equilibrium.
25

Let L (R) denote the set of R-valued adapted processes, and C

missible consumption processes.2 For any c 2 C we let Ct , 0

L (R) the set of ad-

t

T denote the history

of the consumption up to time t : Ct (s) = fc(s ^ t) ; 0

T g. The utility derived

s

from any consumption process is given by a utility function U : C ! R is given by
U (c) = Eu CT :
for some monotonic and concave functional u : D [0; T ] ! R: In contrast to the example
in Section 2.4.1.1, utility is derived from consumption, which is financed through trading
via the budget equation
R
ct + 0 R dt + 0 t dBt ;
t t
t

dXt = Xt rt

X0 = x;

where x 2 R is initial wealth and B is d-dimensional Brownian motion. The instanta-

neously riskless rate r 2 L (R), excess instantaneous expected returns R 2 L (Rn ), and
return diffusion R 2 L Rn n are allowed to be stochastic processes. For simplic-

R
ity, we assume n = d and that t is invertible for all t (therefore markets are complete).
Consider a consumption process given by
dct = c (t; Ct ) dt + c (t; Ct ) dBt ;
where c : [0; T ]

D [0; T ] ! R and c : [0; T ]

existence of a square integrable process

(2.16)

D [0; T ] ! Rd . We suppose the

2 L R++ (the utility gradient density at c)

which satisfies

lim
#0

U (c + h)

U (c)

=E

Z T
0

s hs ds; for all h such that c + h 2 C for some

2 Typically C = L (R) or C = L R
++ .
26

> 0:

(An example is provided below) Finally, suppose

is some functional of the history of the

consumption process:
t = F (t; Ct ) ;

t 2 [0; T ] ;

(2.17)

where F is some smooth strictly positive functional F : [0; T ]

D [0; T ] ! R++ . Then

assuming no constraints on trading, if c in (2:16) is the optimal consumption process, then
is also a state-price density process (see, for example, [12]) satisfying
d t
=
t
where

rt dt

0
t dBt ;

is the market price of risk process defined by t =

R
t

1 R
t .

Applying

functional Itô’s lemma to (2:17) yields the interest rate and market price of risk process:

0

1
1 F (t; C ; [t; T ]) c (t; C )
t
t C
B Dt F (t; Ct ; [t; T ]) + D
@
A
1
+ 2 D2 F (t; Ct ; [t; T ]) k c (t; Ct )k2
;
F (t; Ct )
D1 F (t; Ct ; [t; T ]) c
(t; Ct ) :
F (t; Ct )

rt =
t=

The following example illustrates a utility function class and corresponding utility gradient
density expression.

Example 2.1. The generalized recursive utility process U (see [34]) is part of the solution
(U; ) to the backward stochastic differential equation (BSDE)

dUt =
where G : [0; T ]

R

R

G (t; ct ; Ut ; t ) dt + 0 dBt ;
t

UT = 0;

(2.18)

Rd ! R is assumed differentiable, and concave and Lipschitz

continuous in (ct ; Ut ; t ). Then [38] show that a unique solution exists. The utility of
27

consumption process c 2 C is the initial value U0 of the solution.
smooth functional H : [0; T ]

Suppose for some

D [0; T ] ! R satisfies

1
0 = D1 H (t; Ct ; [t; T ]) c (t; Ct ) + D2 H (t; Ct ; [t; T ])
2
+ G t; ct ; H (t; Ct ) ; D1 H (t; Ct ; [t; T ]) c (t; Ct )

c (t; C ) 2
t

0 = H T; CT :
Then H (t; Ct ) ; D1 H (t; Ct ; [t; T ]) c (t; Ct ) solves the BSDE (2:18) : Using the abbreviation G (t) = G t; ct ; H (t; Ct ) ; D1 H (t; Ct ; [t; T ]) c (t; Ct ) the utility gradient
density at c is the solution of the SDE

t = Et Gc (t) ; where

dEt
= GU (t) dt + G (t)0 dBt ;
Et

2.5

E0 = 1:

Comparison with Dupire’s setup

The following concepts are used in Dupire’s work for the one-dimensional (d = 1) case.
[
t ; where
t2[0;T ]
e
t = D([0; t]; R). Let X(t) 2 R; 0 t T: Derivatives of a functional are based on the
n
o
e
following perturbations of xt = X(s); 0 s t 2 t :
The state space of the process of interest is taken as the vector bundle

xh (s) = xt (s) + h1fs=tg ;
t

s

t;

xt;h (s) = xt (s)1fs<tg + xt (t)1ft s t+hg ;
The distance in space

=

s

t+h

T:

is defined, for x; y 2 ; where xt 2 t and ys 2 s ; as

d (xt ; ys ) = xt;s t

ys
28

1

+s

t;

s

t:

For any functional f :

! R, xt 2 t , the vertical first and second derivatives are defined

as

x f (xt ) = lim
h!0

f (xh )
t

f (xt )
h

;

h
x f (xt )

xx f (xt ) = lim
h!0

x f (xt )
h

;

and the horizontal derivative is defined as

t f (xt ) =

lim
h!0+

f (xt;h )

f (xt )

h

:

Using the above definitions of derivatives and examining the Taylor expansion, Dupire
derived the functional Itô formula for the continuous case.
To see the correspondence between Dupire’s setup and ours, we map the functional f :
e
! R to a functional F : D([0; T ]; R2 ) ! R by defining for each RCLL (X(t); Z(t)) 2

R2 :

F (x; z) = f (xjZ(T )j^T ):
In particular if we take Z(s) = t ^ s, 0

s

T with t 2 [0; T ]; then F (x; z) = f (xt ):

e
It follows that for X(t) = (X(t); t) 2 R2 we get F (Yt ) = f (xt ); where, consistent with
our earlier notation, Yt (s) = X(t ^ s): We also get f (xh ) = F (Yt + h1[t;T ] e1 ) and
t
e
f (xt;h ) = F (X(t ^ ); (t + h) ^ ) = F (Yt + h1[t;T ] e2 ): We conclude that
1
x f (xt ) = D1 (Yt ; [t; T ]);

1
t f (xt ) = D2 (Yt ; [t; T ]):

Finally observe that if f (xt;h ) = f (xt ) 8 0

t

T; 0

h

T

t; then F above

can be considered as a functional on D([0; T ]; R): This will be the case with examples
like the maximum functional, F (x) =
e
max X(t)
jt sj<

e
X(s) :

e
max X(t) or -modulus of continuity, F (x) =
0 t T

29

3
Some preliminary concepts from
Optimization and Financial Economics

3.1

Introduction to Optimization under constraints

This section deals with the methods used in solving constrained optimization problems.
Here, the main goal is to prove variations of Kuhn-Tucker theorem which is a Langrage
multiplier type of result used by us in infinite dimensional setting. This cover problems
where there are multiple constraints that may be non-binding, namely contraints which are
formulated as inequalities.
30

The Kuhn-Tucker conditions are simply the first-order conditions for a constrained optimization problem. Linear programming is a special case covered by the Kuhn-Tucker
theory.
In this section we will be proving variations of Kuhn-Tucker theorem. The method of
the proof is similar to the proof of Kuhn-Tucker theorem in [36](See theorem 1 in Section
9.4). We will start by proving the theorem for general normed spaces and then specialize
to collection of processes. We will need the following definitions for the theorem.
Definition 3.1. Let B and Z be normed spaces, X

B , G : X ! Z and f : X ! R.

a) X is called “extended convex” if 8 x1 ; x2 2 X there is = (x1; x2 ) > 0 so that
x1 + (1
b) Denote Hx

)x2 2 X

1+ .

for all

fh 2 B : x + h 2 Xg; x 2 X. Let X be extended convex and let

x 2 X; h 2 Hx . We denote the Gateaux derivative of G in the direction h as
G(x; h) = lim
!0

G(x + h)

G(x)

:

Observe that since X is extended convex, we don’t have to restrict

to be positive in

the definition of G(x; h). If G(x; h) exists for all h 2 Hx we say that G is Gateaux
differentiable at x.
c) The functional f (x; ) : Hx ! R will be called the supergradient for f at x if
f (x + h)

f (x)

f (x; h);

8h 2 Hx :

d) Let Z contain a convex cone A; namely, x + y 2 A; for each ;
Also we denote x

y (respectively x > y) if x

y 2 A (respectively x

> 0; x; y 2 A:
y 2interior(A))

and A is a positive cone. The point x0 2 X is said to be a regular point of fG(x)
0g if G(x0 )

0 and there is h 2 Hx0 so that G(x0 ) + G(x0 ; h) < 0; where G(x0 ; h) is
31

the Gateaux derivative of G at x0 . Finally, we denote z
dual space of Z) if z [x]

0 for any z 2 Z (Z is the

0 for each x 2 A:

Remark 3.1. (a) Hx is convex (respectively extended convex) if X is convex (respectively
extended convex), but it isn’t necessarily a linear subspace of B. For a mapping T : Hx !
Y , where Y is a vector space, to be linear simply means

T (ah1 + bh2 ) = aT (h1 ) + bT (h2 );

for all a; b 2 R; h1 ; h2 ; ah1 + bh2 2 Hx :

If G(x; ) (respectively f (x; )) is linear in this sense we say that G has a linear Gateaux
derivative (respectively linear supergradient) at x.

(b) We can define the supergradient of G in the same way as we defined for f , and we
can define x0 2 X to be a regular point in the same way as in Definition 3.1(d) with the
understanding that G(x0 ; h) stands for the supergradient of G:
Next we will define a weaker concept than extended convex that will be useful in certain
cases (see the concept of extended convex for processes in Definition 3.3)

Definition 3.2. We will say that X, a subset of a vector space B, is "weakly extended
convex with respect to a collection of functions F = ff : B ! Rd g" if f (X) is extended
convex in Rd for each f 2 F:

Example 3.1 (Extended Convex). Let B be a vector space whose elements are all the realvalued functions defined on the set A (say). Let C = ff 2 B : f (x) > 0; x 2 Ag.
Obviously C

B is convex. A simple condition for C to be extended convex is:

0 < inf ff (x)g
x2A

sup ff (x)g < 1;
x2A
32

for all f 2 C:

3.1.1

Kuhn-Tucker Theorem

Theorem 3.1 (Kuhn-Tucker). Let B be a normed space, Z a normed space that contains
B extended convex, G : X ! Z and

a positive cone P with non-empty interior, X
f : X ! R: Let x0 2 X satisfy G(x0 )

0 and f (x0 ) =

Assume:

min
ff (x)g.
fx2X: G(x) 0g

(i) G has a linear Gateaux derivative and f has a linear supergradient at x0 .
(ii) x0 is a regular point of fG(x)
Then there is z 2 Z ; z

0g.

0 such that

f (x0 ; h) + z [ G(x0 ; h)] = 0; h 2 Hx0 ;
z [G(x0 )] = 0:

Proof. In the space W = R

A = f(r; z) : r

Z, define the sets

f (x0 ; h); z

G(x0 ) + G(x0 ; h) for some h 2 Hx0 g;

D = f(r; z) : r

0; z

0g:

The set D is obviously convex. Letting (r1 ; z1 ) and (r2 ; z2 ) 2 A, and using the fact that
the supergradient of f and the Gateaux derivative of G are linear, we have ( r1 + (1
)r2 ; z1 + (1

)z2 ) 2 A where 0

1. Therefore A is also convex. The set D

contains interior points because P does. If (r; z) 2 A, with r < 0 and z < 0, then there
exists h 2 Hx0 such that
f (x0 ; h) < 0;

G(x0 ) + G(x0 ; h) < 0:

The point G(x0 ) + G(x0 ; h) is the center of some sphere of radius
negative cone in Z. Then for 0 <

contained in the

< 1 the point (G(x0 ) + G(x0 ; h)) is the center
33

of an open sphere of radius

contained in negative cone; hence so is the point (1

)G(x0 ) + (G(x0 ) + G(x0 ; h)) = G(x0 ) +

G(x0 ; h).

By the definition of the

Gateaux derivative we have

k G(x0 + h)

G(x0 ; h) k= o( ):

G(x0 )

Therefore G(x0 + h) < 0 for sufficiently small . By definition of supergradient for
f we have f (x0 + h)

f (x0 )

f (x0 ; h). Then the supposition f (x0 ; h) < 0 implies

f (x0 + h) < f (x0 ). This contradicts the optimality of x0 ; therefore A contains no interior
points of D.

By Theorem 3, Section 5.12 (from [36]), there is a hyperplane separating A and D.
Hence there are r0 , z , such that
r0 r + z [z]

for all (r; z) 2 A;

r0 r + z [z]

for all (r; z) 2 D:

Because (0; 0) belongs to both A (choose h = 0) and D, we have
once that r0

0 and z

= 0. It follows at

0 (otherwise, you can choose (r; z) 2 D with one component 0

and the other negative, resulting in a contradiction). Furthermore, r0 > 0 because of the
existence of h 2 Hx0 such that G(x0 ) + G(x0 ; h) < 0 (by regularity of x0 ). By scaling
we can assume without loss of generality that r0 = 1. From the separation property, we
have for all h 2 Hx0
f (x0 ; h) + z [G(x0 ) + G(x0 ; h)]

0

(because trivially ( f (x0 ; h); [G(x0 )+ G(x0 ; h)]) 2 A). Setting h = 0 gives z [G(x0 )]
0 but G(x0 )

0; z

0 implies z [G(x0 )]
34

0 and hence z [G(x0 )] = 0. We conclude

that
f (x0 ; h) + z [ G(x0 ; h)]

0;

8h 2 Hx0 :

For any h 2 Hx0 , extended convexity of X implies that there exists
h 2 X; that is,

x0

(3.1)
> 0 such that

h 2 Hx0 : By linearity of the supergradient and Gateaux

derivative with respect to h, we have

f (x0 ; h) + z [ G(x0 ; h)] = f (x0 ;

h) + z [ G(x0 ;

h)]:

The last equation together with equation 3.1 gives

f (x0 ; h) + z [ G(x0 ; h)] = 0.

Corollary 3.1. Using a proof similar to that of Theorem 3.1, we can replace assumption
(i) of that theorem by one of the following assumptions
a) G has a linear Gateaux derivative and f has a linear Gateaux derivative at x0 .
b) G has a linear supergradient and f has a linear Gateaux derivative at x0 :
c) G has a linear supergradient and f has a linear supergradient at x0 :
Next we will prove below a corollary to Theorem 3.1 for processes. We will start with
a variation of the definition of extended convex which is appropriate to a collection of
stochastic processes.
Definition 3.3. A collection of vector-valued stochastic processes X will be called extended
convex if for all x1 ; x2 2 X there is a process
process

= (!; t) that satisfies

= (!; t; x1; x2 ) > 0 such that for each

1 + we have

)x2 2 X:

x1 + (1
35

Observe that the concept just defined is essentially a special case of weakly extended convex
[0; T ]. Namely, each (!; t) represents

(see Definition 3.2) with respect to the functionals

a functional given by: x ! x(!; t), for each process x 2 X.
n
Example 3.2. To show that L2 Rn
++ is extended convex, for any x1 ; x2 2 L2 R++
we let (t) = min (x1 (t) ; x2 (t)) = max (x1 (t) ; x2 (t)).
We next define the concepts of supergradient density and gradient density for realvalued functionals defined on a collection of processes.

Definition 3.4. Let X

L2 (Rn ) be an extended convex subset of processes. Let

: X!R

be a functional. For any x0 2 X;
(a) The process

2 L2 (Rn ) is a supergradient density of
(x0 + h)

(b)

( jh)

(x0 )

2 L2 (Rn ) is a gradient density of
( jh) = lim
#0

Corollary 3.2. Let X

at x0 if

for all h 2 Hx0 .

at x0 if

(x0 + h)

(x0 )

for all h 2 Hx0 .

L2 (Rn ) be an extended convex set of processes. Let G : X !

Rm and f : X ! R: Let x0 2 X satisfy G(x0 )
Assume:

0 and f (x0 ) =

min
ff (x)g.
fx2X: G(x) 0g

(i) G has a gradient density, and f has a supergradient density or a gradient density at
x0 :
(ii) x0 is a regular point of fG(x)

0g.
36

Then there is a z 2 Rm such that
+
f (x0 ; h) + z 0 [ G(x0 ; h)] = 0;

all h 2 Hx0 ;

z 0 [G(x0 )] = 0:
Proof. Recall that all the processes that we deal with are assumed to be progressively
measurable. We will prove the corollary under the assumption that f has a supergradient density at x0 : The proof goes exactly as Theorem 3.1 through equation (3:1). Fix
some h 2 Hx0 : Because X is extended convex there exists a

2 L R++ such that

h 2 X. For each " > 0; define A(") = f(!; t) : (!; t) > "g. Because 1A is a

x0

progressively measurable process and "1A(") < ; we have x0

"h1A(") 2 X: As in the

proof of Theorem 3.1 we get (from equation (3:1))

f (x0 ; h1A(") ) + z

0h

i
G(x0 ; h1A(") ) = 0

Since gradient and supergradient densities are continuous in the increment and h1A(") !

h in L2 (Rn ) as " ! 0; we have

f (x0 ; h) + z 0 [ G(x0 ; h)] = 0:
The proof is similar when we assume that f has gradient density at x0 :

3.2

Generalized Recursive Utility

In this section, we define recursive utility. We also present computations of supergradient
density and gradient density for recursive utility. Continuous-time recursive utility was first
defined in [13]. The generalized recursive utility class was introduced in [34] to unify the
stochastic differential utility (SDU) formulation of [13], and the multiple-prior formulation
37

of [5]. Generalized recursive utility is defined as a solution to a general BSDE.

3.2.1

Recursive utility and BSDEs

All uncertainty is generated by d-dimensional standard Brownian motion B over the finite
time horizon [0; T ], supported by a probability space ( ; F; P ). All processes appearing
in this paper are assumed to be progressively measurable with respect to the augmented
filtration fFt : t 2 [0; T ]g generated by B. For any subset V
Rn (respectively V
Rn m ), let L (V ) denote the set of V -valued progressively measurable processes, and, for
any p

1, denote

Lp (V ) =

(

x 2 L (V ) : E
8
<

"Z

T

0

#

)

kxt kp dt + xT p < 1 ;

9
=
4 esssup kxt kp 5 < 1 ;
Sp (V ) = x 2 L (V ) : E
:
;
t2[0;T ]
2

3

where kxt k2 = x0 xt (respectively trace(x0 xt )). It is well known that L2 (V ) with V
t
t
n is an inner product space with inner product defined by
R

(x jy ) = E

"Z

T

0

#

x0 yt dt + x0 yT ;
t
T

x; y 2 L2 (V ) :

The qualification “almost surely” is omitted throughout.
We have N
C

L2 Rk , k

1 agents.

The set of consumption plans is an extended convex set

N (see Definition 3.3 in Appendix B for a precise formulation of

extended convex set of processes). For any c 2 C, we interpret ct , t < T , as a length-k
vector of consumption rates, and cT as a vector of lump-sum terminal consumption. For
any c 2 C and bounded b 2 L(Rk ) we assume that c 2 C where ci = max ci + bi ; ci =2 ,
~
~t
t
t t
i = 1; : : : ; k, t 2 [0; T ].
38

Remark 3.2. An example of a set C that satisfies the required assumptions is the following:
C = fc(t; !) 2 At;! ; 0

t

T; ! 2 g \ L2 Rk ;

where At;! is a convex and open set in Rk containing Rk .
++
For any c 2 C we define the RN -valued utility process Z as part of the pair (Z; ) 2

L2 R N

L2 RN d that solves the BSDE
dZt (c) =

The function
P

:

(t; ct ; Zt ; t ) dt + t dBt ;

[0; T ]

Rk

RN

ZT =

T; cT :

(3.2)

RN d ! RN is called the aggregator and is

B(Rk+N +N d ) measurable, where P is the predictable -field on

[0; T ]. The

terminal-utility aggregator does not depend on ZT ; T . We let i (t) and i denote the
t
th row of (t) and (t), respectively. Let
N k denote the matrix with typical
i
c2R
ij
element c = @ i =@cj ;
2 RN the vector with typical element i j = @ i =@Z j ;
Zj
Z
ij
and
2 RN d the matrix with typical element
= @ i =@ mj .
m
m
Initial BSDE existence and uniquness results, based on the type of Lipschitz continuity
condition assumptions were first obtained in [38]. Based on the same assumptions [17]
gave an better proof.

BSDE theory has been further developed in [23], [32], [4] and

others(see [31]). We will be using Lipschitz-assumptions for existence and uniqueness
in this section but in later chapters, we will assume weaker conditions like quadratic or
concave aggregators.
Condition 3.1. a) i (t; ) ; i = 1; : : : ; N has continuous and uniformly bounded derivatives w.r.t. c; Z; .
b) i (t) > 0 for all t 2 [0; T ] and i = 1; : : : ; N .
c
c) (t; 0; 0; 0) 2 L2 RN :
d) E( (T; cT ) 2 ) < 1:
39

Remark 3.3. Under Condition 3:1, there exists a unique pair (Zt ; t ) 2 L2 RN
L2 RN d which solves (3:2) for each c 2 C:

3.2.2

Utility supergradient and gradient density calculation

Consider a solution (Z; ) to (3:2) for c 2 C. Define the adjoint process ", with some
initial value "0 2 RN n f0g, as the solution to the SDE (see the notation for derivatives in
+
Section 3.2.1)

d"i =
t

N
N
X j j
X j j
0
"t
(t; ct ; Zt ; t ) dt +
"t
i
i (t; ct ; Zt ; t ) dBt ;
Z
j=1
j=1

i = 1; : : : ; N ;
(3.3)

which can be written more compactly as
0
d"i = "0
t
t Z i (t; ct ; Zt ; t ) dt + "t

i (t; ct ; Zt ; t ) dBt ;

i = 1; : : : ; N:

Observe that condition 3.1(a) imply existence and uniqueness of the adjoint process ".
We will be proving two lemmas which will involve the calculation of super gradient
and gradient density.
Lemma 3.1. Suppose c 2 C and (Z; ) solve the BSDE (3:2) and " solves the SDE (3:3)
j
j
for some "0 2 RN . Assume i (t; ) is a concave function,
i (t) 0 and
i (t) = 0
Z
for all i 6= j, i; j 2 f1; 2; : : : ; N g and t 2 [0; T ] : Then for all h such that c + h 2 C we
have
"0 fZ0 (c + h)
0

Z0 (c)g

E

Z T
0

"0 c (t; ct ; Zt ; t )ht dt + "0 c (T; cT )hT
t
T

!

.

We will first start by proving the positivity of "t ; solution to SDE (3:3) :
Lemma 3.2. Assume

j
(t)
Zi

0 and

j

i (t) = 0 for all t 2 [0; T ], i 6= j , i; j 2

40

f1; 2; : : : ; N g : Then "i ; the solution of equation (3:3), satisfies "i
t
t
1; : : : ; N; 0

t

0; if "i
0

0; i =

T:

Proof. Consider the SDE
i
i
i
d t = t i i (t; ct ; Zt ; t ) dt + t i i (t; ct ; Zt ; t )0 dBt ;
Z

i = 1; : : : ; N:

Under Condition 3.1 and the assumptions for the lemma, we can easily check that the
conditions of Theorem 1.1 of [21] are satisfied. It follows from their theorem that "i
0
implies "i
t

i
i
i
t ; i = 1; : : : ; N: Therefore, by selecting "0 = 0
"i
t

i
t

0;

t 2 [0; T ] ;

i
0

0;we get

i = 1; : : : ; N:

Using the nonnegativity of ", we now prove Lemma 3.1. Let c; c + h 2 C, and let
(Z (c) ;

(c)) and (Z (c + h) ;

= Z (c + h)

Z (c) and =

d "0 t =
t

"0
t

8
>
<
>
:

(c + h)) denote the solutions to equation (3:2). Define
(c + h)

(c). Using integration by parts we have

Z (t; ct + ht ; Zt +
(t; ct ; Zt ; t )

t; t + t)

Z (t) Zt

9
>
=

;
(t) ; t >

dt + Mt

for some local martingale M , where we use the abbreviation for derivatives

i (t) =

i (t; ct ; Zt ; t ) for i 2 fc; Z; g. By concavity,
(t; ct + ht ; Zt +

t; t + t)

(t; ct ; Zt ; t )

c (t) ht

Z (t) Zt

(t) ; t

Let f n : n = 1; 2; : : :g be an increasing sequence of stopping times such that n ! T ,
a.s., and M stopped at n , i.e., Mt^ n , is a martingale. Integrating and taking expectation
41

0:

we get (note that the " is nonnegative by Lemma 3.2)
"0 0
0

E

Z

0

n 0
"t c (t) ht dt + "0 n

n :
h

i2
Because Z (t) and
(t) are uniformly bounded we get E supt T k"t k < 1. Ush
i2
ing a similar argument as in Proposition 2.1 from [17] we get E supt T kZt k < 1:
h
i
Therefore E supt T k"t k
supt T kZt k < 1: Letting n ! 1 and interchanging limit and expectation we get
"0 0
0

E

Z T
0

"0 c (t) ht dt + "0 c (T ) hT
t
T

!

using also the concavity of Z i (T ) = i (T; cT ) in cT for each i.

Lemma 3.3. Suppose c 2 C and (Z; ) solve the BSDE (3:2).

For any initial value

"0 2 RN , the utility gradient density of "0 Z0 at c is given by
0
0
t = "t c (t; ct ; Zt ; t );

t < T;

0
T = "T c (T; cT );

where " satisfies the SDE (3:3).
2
We start by defining the following notation. Given x 2 Rn d and y 2 Rn , we define
the product between x and y as
n

o
0 y ; i = 1; : : : ; N :
(x; y) = trace x
Proposition 3.1 (linear BSDE). Let

2 L Rn n and

2 L Rd both be uniformly

bounded, ' 2 L2 (Rn ), and E(k k2 ) < 1. Then the linear BSDE
dYt = ('t + t Yt + Zt t ) dt
42

Zt dBt ;

YT = ;

(3.4)

has a unique solution (Y; Z) in L2 (Rn )

L2 Rn d . Furthermore Yt satisfies
Z T

0 +
T
t

0Y = E
t
t t

!

0 ' ds ;
s s

(3.5)

where t is the Rn -dimensional adjoint process defined by the forward linear SDE
0
d t = t t dt + t t dBt ;
Proof. Since

and

n
0=a2R :

(3.6)

are bounded processes, the linear aggregator is uniformly Lipschitz,

and therefore there exists a unique solution to BSDE 3.4. Applying Ito’s lemma we get

d

0Y +
t t

Z t
0

0 ' ds
s s

!

= (d t )0 Yt + 0 dYt + (d t )0 dYt
t
0
= ( t t dBt )0 Yt + 0 Zt dBt ;
t

which is a local martingale. Theorem 2.1 of [17] implies Y 2 L2 (Rn ). By a similar
argument as in Proposition 2.1 from [17] we have sup kYs k and sup k s k are squares T
!
! s T

sup kYs k
sup k s k is an integrable random
s T
s T
Therefore the local martingale is a martingale, and so integrating and taking

integrable random variable and
variable.

conditional expectation with respect to Ft we get (3:5).
Remark 3.4. For every t and 1

i

n there exists an initial value a 2 Rn such that t =

ei and therefore
Yti = Et

Z T

0 +
T
t

!

0 ' ds :
s s

This will follow from the representation of solution of the SDE (3:6) as t = At a; where
At is a nonsingular n

n matrix.

Next we present a version of Proposition 3.1 that will be useful in the proof of Lemma
43

3.3.
Proposition 3.2. Let

2 L Rn n and

2
2 L Rn d

both be uniformly bounded,

' 2 L2 (Rn ), and E(k k2 ) < 1. Then the BSDE

dYt = ('t + t Yt + ( t ; Zt )) dt
has a unique solution (Y; Z) in L2 (Rn )
0Y = E
t
t t

Zt dBt ;

YT = ;

L2 Rn d . Furthermore Yt satisfies
Z T

0 +
T
t

!

0 ' ds ;
s s

where, for any a 2 Rn , t is the Rn -dimensional adjoint process defined by the forward
linear SDE
d i=
t
and where i (t) is the n

n
n
X j ji
X j j
t t dt +
t i (t)dBt ;
j=1
j=1

0 = a;

d matrix defined by concatenating the ith row from each n

d

block in t (consisting of n blocks).
Proof. The proof is similar to Proposition 3.1.
Proof of Lemma 3.3. Let h be a process such that c + h 2 C: Since C is (extended) convex,
we have c + h 2 C for any constant

2 [0; 1]. Let (Z ;

) be the solution to the BSDE

(3:2) corresponding to c + h. By the results on BSDEs in [3], the derivative (@Z; @ ) of
(Z ;

) with respect to

is given by the solution of following BSDE:

d@Zt = ( c (t)ht + Z (t)@Zt +(

(t); @ t ))dt (@ t )dBt ;

@ZT = c (T; cT )hT :
(3.7)

To get the exact form of (@Z; @ ), we use the adjoint process "t 2 RN presented as a solution of the equation (3:3). Observe that due to Condition 3.1 we have existence and uniqueh
i
ness of ". By Proposition 3.2 (Observe that in [3] we have E esssupt2[0;T ] kZt k2 < 1
44

and equation (3:3) is a linear SDE), the solution (@Z; @ ) of equation (3:7) is given by
"0 @Zt = Et
t

Z T
t

"0 c (s; cs ; Zs ; s )hs + "0 c (T; cT )hT
s
T

45

!

;

t 2 [0; T ] :

4
General Maximization Principle

4.1

Introduction

We study a class of optimization problems involving linked recursive preferences in a
continuous-time Brownian setting. Such links can arise when preferences depend directly
on the level or volatility of wealth, in principal-agent (optimal compensation) problems
with moral hazard, and when agents are altruistic in the sense that utility is affected by the
utility and risk levels of other agents. We characterize the necessary first-order conditions,
which are also sufficient under additional conditions ensuring concavity. We also examine applications to optimal consumption and portfolio choice, and applications to Pareto
46

optimal allocations.
The optimization problems we study all reduce to maximizing a linear combination of
a multidimensional BSDE system. This system was proposed in [17] as an extension of
[13]’s stochastic differential utility (SDU). [34] show, in the single-agent (one-dimensional)
case, that this recursive specification allows considerable flexibility in separately modeling
risk aversion and intertemporal substitution, and unifies many preference classes including
SDU and multiple-prior formulations ([5]; [2]; [37]). We show that the multidimensional
analog can be used to model altruism and direct dependence of utility on wealth, as well as
the links in utility induced by moral hazard in principal/agent problems (see [35]).
Another contribution of our paper is that we define a multidimensional extension of
the translation-invariant (TI) class of BSDEs, introduced by [44] as an extension of timeadditive exponential utility. We show that the solution to this class of BSDEs can simplify to the solution of a single unlinked BSDE and a system of pure forward equations.
Furthermore the solution method simplifies and easily generalizes, and the conditions for
sufficiency are relaxed compared to the general case. The simplification of the solution
for this class is illustrated in Example 4.9 which solves the optimal consumption/portfolio
problem with homothetic preferences and direct utility for wealth.
Our solution method is based on an extension of the utility-gradient approach originating in [8] and [30] for additive utilities, and extended by [46] and [15] to recursive
preferences. Our general optimization result (Theorem 4.1 below) can be viewed as the
natural multidimensional extension of Theorem 4.2 of [18]. They derive a maximum principle for the optimal consumption/portfolio problem of a single agent with recursive utility
and nonlinear wealth dynamics. They formulate their problem in terms of BSDEs for utility and wealth and obtain first-order conditions (FOCs) in terms of two adjoint processes,
which represent utility and wealth gradient densities. We consider a general system of
linked BSDEs and obtain FOCs in terms of a system of linked adjoint processes.
47

4.2
4.2.1

General Maximization Principle
Optimization Problem

We will use the same setting as Chapter 3. The definition of utility and the condition 3.1
from Chapter 3 will be assumed in this chapter. We have N agents and their preferences
follow generalized recursive utility.
Fixing some nonzero weights

2 RN , the problem is
+

max 0 Z0 (c) subject to Z0 (c)
c2C
i
(i.e., Z0 (c)

K

(4.1)

K i , i = 1; : : : ; N ), where K i 2 R [ 1 (to allow for void constraints).

Next we present the maximum principle for multidimensional BSDEs, which is our
solution to (4:1).
Theorem 4.1 (Maximum Principle). Suppose c 2 C and (Z; ) solve the BSDE (3:2) and
" solves the SDE (3:3).
a) (Necessity) If c solves the problem (4:1) then there is some

"0 =

+ ;

"0 c (t; ct ; Zt ; t ) = 0;
t

0 fZ (c)
0

Kg = 0;

Z0 (c)

2 RN such that
+

t 2 [0; T ] ;

(4.2)

K:

j
j
b) (Sufficiency) Assume i (t; ) is a concave function,
i (t) 0 and
i (t) = 0 for all
Z
i 6= j, i; j 2 f1; 2; : : : ; N g, and t 2 [0; T ] : If (4:2) holds then c is optimal.
Proof. a) We will use Corollary 3.2 from Chapter 3 with

G = Z0 (c)

K; and

f =

0 Z (c) to prove the necessity part. By Definition 3.1(d) in Chapter 3, c is a regular point
0
of fc 2 C : G(c)

0g if G(c) + G(c; h) < 0 for some h such that c + h 2 C, and where

G(c; h) is the Gateaux derivative of G at c in the direction h 2 C. By Lemma 3.3, we
see that in our case the condition for a regular point is satisfied if for every initial value
48

"0 = ei , i = 1; : : : ; N , there exists an h with c + h 2 C and
Z T

E

0

"0 c (t; ct ; Zt ; t )ht dt + "0 c (T; cT )hT
t
T

!

> 0:

(4.3)

Obviously since "0 = ei ; it follows that the solution " of the SDE (3:3) is not identically
zero. From the assumed properties of C it follows that c + h 2 C for h defined by
o
n
ci =2 _ "0 i (t; ct ; Zt ; t ) ;
t c
t

hi = 1 ^
t

i = 1; : : : ; k,

t 2 [0; T ] :

(4.4)

It is easy to confirm (4:3) for the h defined above. By Corollary 3.2, if c is optimal then
2 RN such that the Gateaux derivative of 0 Z0 (c) + 0 fZ0 (c) Kg in
+
the direction of h is 0 for all h such that c+h 2 C and 0 fZ0 (c) Kg = 0. Using Lemma
0K
3.3 to compute the Gateaux derivative, or utility gradient density, for ( + )0 Z0 (c)
there exists a

we therefore get

E

Z T
0

"0 c (t; ct ; Zt ; t )ht dt + "0 c (T; cT )hT
t
T

!

= 0;

t 2 [0; T ] ;

8h such that c + h 2 C;
where " solves the SDE (3:3) with initial value "0 =

+ .

Because the above statement is true 8h such that c+h 2 C; we have "0 c (t; ct ; Zt ; t ) =
t
0, t 2 [0; T ] (otherwise we get a contradiction using h defined as in (4:4)). This completes
the necessity proof.
b) Lemma 3:1, "0 =
0 fZ (c + h)
0

+

Z0 (c)g

and (4:2) together imply
0 fZ (c)
0

Z0 (c + h)g = 0 fK

Z0 (c + h)g

(4.5)

(the equality follows from complementary slackness).
It follows immediately that 0 Z0 (c + h) > 0 Z0 (c) implies 0 fK
49

Z0 (c + h)g >

0, and therefore a violation of at least one constraint.
We now use Theorem 4:1 to sketch a characterization of the optimum as the solution
to an forward-backward stochastic differential equation (FBSDE) system. Furthermore, a
reduction in dimensionality is attained, which can be useful for small N .
Assume that "i is strictly positive for some i 2 f1; 2; : : : ; N g; by relabeling we obtain

strict positivity of "1 . Define t by

t=

"2
"N
1; t ; : : : ; t
"1
"1
t
t

!0

t 2 [0; T ] :

;

(4.6)

The FOCs (4:2) imply 0 c (t; ct ; Zt ; t ) = 0. Assume that we can invert the FOCs to
t
solve for consumption. That is, there exists a function ' :
[0; T ] RN RN d ! Rk

satisfying 0 c (t; ' (t; t ; Zt ; t ) ; Zt ; t ) = 0. Applying Ito’s lemma to compute the
t
dynamics of i , we can express the FOCs (4:2) as a FBSDE system for (Z; ; ):

dZt (c) =

(t; ct ; Zt ; t ) dt + t dBt ; ZT =
T; cT ;
n
i
i
0 (t)
d i= 0
(t)
t
t
t Z 1 (t) + t 1 (t)
1
Zi
+ 0
t
i =
0

n

i (t)

i+ i =

i
t

0 (t)
i (t)
1

o

t dt
(4.7)

o
(t) dBt ;
1

1+ 1 ;

i = 2; : : : ; N;

ct = ' (t; t ; Zt ; t ) ;
0 = 0 fZ0 (c)

4.3

Kg ;

Z0 (c)

K;

0:

Translation Invariant (TI) BSDEs

In this section we examine an aggregator form which leads to a particularly tractable solution. We show that when k = N

1 the solution reduces to solving a single unlinked
50

backward equation, followed by a system of N

1 forward SDEs. Furthermore, we use a

dynamic-programming argument that relaxes the sufficiency conditions of Theorem 4.1.
We assume throughout this section that C satisfies, in addition to the previously assumed

properties, the following: c + v 2 C for all c 2 C and v 2 Rk . We also assume a TI
.
aggregator, defined as follows:
Definition 4.1. A TI aggregator takes the form

(t; c; Z; ) =

for some

:

[0; T ]

rank(M ) = k, and

=

We can interpret

(t; M c

Z; ) ;

(T; c) = M c + ;

RN

RN d ! RN , where M 2 RN k is assumed to satisfy
1 ; : : : ; N 0 2 F is assumed to satisfy E k k2 < 1.
T

as supplemental lump-sum terminal consumption in addition to ter-

minal component of the control c (say, a lump-sum endowment; an intermediate endowment can be included via the ! argument of ).
A key property of the TI class is the (easily verified) quasilinear property

Zt (c + ) = Zt (c) + M ;

t 2 [0; T ] ;

Special cases of the TI class are common in finance.

for all

2 Rk .

(4.8)

Example 4.9 shows that ho-

mothetic preferences and the standard present value operator (that is, the budget equation
considered as a BSDE) are both within the TI class after transforming to logs. The following example shows that additive exponential preferences are also in the TI class.
Example 4.1. Suppose
i (t; M c

Z; ) =

exp

ai0 (M ct

Zt )

1 i0 i
;
2 t t

i = 1; : : : ; N;

0
N
i
where ai = ai ; : : : ai
1
N 26 R and we normalize with ai = 1 for i = 1; : : : ; N . Then
51

i
the ordinally equivalent utility zt =

i
zt =

8
<Z T

Et
:

t

exp

i
Zt satisfies (under sufficient integrability)

exp

ai0 M cs

Y

i
j a
zs j ds + exp

M i cT + i

j6=i

9
=
;

:

Agent i’s utility is increasing (decreasing) in agent j’s utility if ai < 0 (ai > 0). If ai = ei
j
j
then agent i’s utility is standard additive exponential.
We show below that the matrix M plays a key role in determining the properties of
the solution. A specification that arises in principal-agent problems and Pareto efficiency
problems is given in the following example.
Example 4.2. Suppose each agent’s aggregator depends only their own consumption and
utility level, and total consumption is given by some stochastic process C 2 L (R). Defin0
ing ct = c1 ; : : : ; cN 1 as the first N 1 agents’ consumption (and so k = N 1),
t
t
then cN = Ct 10 ct , where 1 denotes a vector of ones. Then M is obtained by stacking a
t
R(N 1) (N 1) identity matrix IN 1 on top of an R(N 1) -valued row vector of 1s:
0

1

B I
C
M =@ N 1 A
10

(4.9)

The agents’ aggregators in this simple case take the form
i
dZt =
N
dZt =

f i t; ci
t

i
Zt ; t dt + i0 dBt ;
t
1 0 ct

f N t; Ct

for some f i :

[0; T ]

R

i
ZT = ci + i ;
T

N
Zt ; t dt + N 0 dBt ;
t
RN d ! R.

i = 1; : : : ; N

N
ZT = Ct

1;

10 cT + N ;

For example, in a class of principal-

agent problems with a single principal (let agent-N represent the principal here) and agents
1; : : : ; N 1, the principal’s problem is to choose the pay processes c to maximize the utility
of the principal’s (i.e.,

= eN ) cash-flow process C
52

10 c, which is what remains of C

i
after the agents are paid, subject to the "participation constraint" Z0
(and K N =

K i , i = 1; : : : ; N 1

1). The agents’ cumulative actions/effort changes the measure, which adds

a dependence of each aggregator on the other agents’ utility diffusion processes.
The key to the tractability of the TI class is the next lemma which shows that the adjoint
processes " must lie in the null space of M 0 . When k = N

1, this implies that

defined

in (4:6) is a constant vector.
Lemma 4.1. In the TI class, at the optimum c we have
"0 M = 0;
t
where "t satisfies (3:3) with "0 =

t 2 [0; T ] ;

(4.10)

+ :

Proof. On the one hand, letting "0 =

+ , Theorem 4:1 implies

"0 fZt (c + v)
lim t
#0

Z0 (c)g

= 0:

On the other hand, by quasilinearity (4:8), the left-hand side equals "0 M v.
t
0 M v = 0 for every v 2 Rk and t 2 [0; T ].
"t

Therefore

Lemma 5.2 implies the following necessary condition on M for an optimum to exist,
which we assume throughout the rest of this section.1
Condition 4.1.
v 0 M = 0 for some v

:

(4.11)

1 By Farkas’ lemma, Condition 4.1 rules out the existence of any (Pareto improving)
fixed consumption increment x 2 Rk satisfying M x 2 Rk and 0 M x > 0; that is, by
+
quasilinearity (4:8), an increment that reduces no agent’s utiilty but strictly improves at
least one’s.
53

Remark 4.1. By Condition 4.1 we can choose some v
^
^ = min f

0: v
^

satisfying v 0 M = 0; let
^

g, and define v = ^ v (i.e., we choose the "smallest" v satisfying
^

(4:11)). Supposing a solution to problem (4:1) exists, we get that v i > i implies that ith
constraint is binding. In Example 4.2, with M given by (4:9) and
and the first N

= eN , then v = 1

1 constraints must therefore (be nonvoid and) bind at any solution.

It follows from the next lemma that the existence of a unique solution to problem (4:1)
implies that in the TI class at most N k constraints in (4:1) are nonbinding at the optimum.
Note that the Lemma applies even with void constraints, implying that there cannot be a
unique solution with more than N

k void constraints.

Lemma 4.2. Under the TI aggregators class, if a solution to (4:1) exists, and if more than
N

k constraints are nonbinding, the solution is not unique.

Proof. Let c be a solution to (4:1) and suppose exactly n
^

N constraints in (4:1) are

nonbinding; without loss of generality assume that these correspond to i = 1; : : : ; n. That
i c
i c
is, Zt (^) > K i , i = 1; : : : ; n; and Zt (^) = K i , i = n + 1; : : : ; N .
Decompose
i0
h
M = M a0 ; M b0 where M a 2 Rn k and M b 2 R(N n) k ; similarly, decompose
i
h
i
i
h
i
h
h
a0 ; Z b0 0 , = a0 ; b0 0 , K = K a0 ; K b0 0 and " = "a0 ; "b0 0 . Defining M 2
~
Z= Z
0
0 0
0
1
a0 M a
C
(N n+1) k by M = B
~
~
R
@
A, then nonuniqueness is implied if rank M < k.
b
M
~
This follows because then there is a x 2 Rk satisfying M x = 0, and therefore (using (4:8))
b c
Zt (^ + x)
Choose

b c
a c
a0 Z a (^) =
a0 M a x = 0.
Zt (^) = M b x = 0 and a0 Zt (^ + x)
t c
a c
a c
2 R sufficiently small that Zt (^ + x) = Zt (^) + M a x K a . From "a =
0

a (from (4:2) and the supposition that the first n constraints are nonbinding) we get that

~
"0 M = 0 (which is implied by optimality of c and (4:10)) is equivalent to 1; "b0 M = 0,
^
0
0
~
~
which implies rank M
N n. Therefore n > N k implies rank M < k and
therefore nonuniqueness.
54

The main result of this section is Theorem 4.2 below, which provides a sufficiency
proof together with a method for constructing a solution to the problem (4:1) under TI
aggregators. We first provide a brief sketch of the solution method based on the results of
Theorem 4.1 and Lemma 5.2. But we will see that the conditions required for the proof of
Theorem 4.2 are considerably weaker than the sufficiency conditions for Theorem 4.1.
We assume throughout the rest of this section that k = N

1, in which case a unique

solution implies at most one nonbinding constraint; the key simplification in this case is that
the null space of M 0 , in which "t lies, is one dimensional (that is,

is a constant vector).

In light of Lemma 4.2 we rearrange the equations so that K i 2 R, i = 1; : : : ; N

KN =

1. Therefore if a unique optimum exists, the first N

1, and

1 constraints bind.

Choose v as in Remark 4.1. Defining
Yt = v 0 Zt ;

xt = M ct

Zt ;

t 2 [0; T ];

(4.12)

we have the FOC condition, from (4:2) and (4:10).
v 0 x (t; xt ; t ) M = 0;
Assuming invertibility, the N

t 2 [0; T ]:

1 conditions in (4:13), and the identity Yt =

is implied by the identities in (4:12), together imply

xt =
for some

:

[0; T ]

R

(t; Yt ; t ) ;

RN d ! RN .

55

t 2 [0; T ];

(4.13)
v 0 xt , which

A constant
N
X

vj

j=1

(

j

also implies a zero diffusion term in (4:7):
vi
v1

i (t; (t; Yt ; t ) ; t )

!

j

)

1 (t; (t; Yt ; t ) ; t )

= 0;

(4.14)

i = 2; : : : ; N:
The restrictions (4:14) together with the identity v 0 t = Y 0 can, assuming invertibility,
t
Y for some :
allow us to obtain t =
t; Yt ; t
[0; T ] R Rd ! RN d . We
then solve the BSDE for Y; Y :

dYt =

v0

t;

t; Yt ; Y
t

t; Yt ;

t; Yt ; Y
t

;

dt + Y dBt ;
t

YT = v 0 :
(4.15)

The solution for Y gives us the diffusion coefficients of Z. We solve the forward equations corresponding to the binding constraints:
i
dZt =

i t;

i = 1; : : : ; N

t; Yt ;

t; Y
t

t; Y
t

;

dt + i0 dBt ;
t

i
Z0 = K i ;

1:

For a vector Z (matrix M ), we denote Z ( i) (M ( i) ) to be the vector (matrix) with ith
element (row) removed.

We solve for optimal consumption c from Z ( N ) in (4:18)
^

below.
The solution method is made more transparent and simple in the following theorem.
Theorem 4.2. Suppose, for all t 2 [0; T ],
xt ; ^ t = arg
^

max
v 0 (t; x; )
(x; )2RN RN d

subject to
v0x =

Yt ;

v0

= Y 0;
t
56

(4.16)

and Y; Y

uniquely solves the BSDE

dYt =

v0

t; xt ; ^ t dt + Y 0 dBt ;
^
t

YT = v 0 :

(4.17)

Then the optimal policy is

ct =
^
cT =
^

M
M

( N)

1

( N)

1

( N)
( N)
Zt
+ xt
^
;
( N)
ZT

t 2 [0; T );

(4.18)

( N) ;

where Z ( N ) solves the forward SDE system
i
dZt =

i t; x ; ^ dt + ^ i dB ;
^t t
t t

i
Z0 = K i ;

i = 1; : : : ; N

Furthermore, the optimal objective function is 0 Z0 (^) = Y0
c

(v

1:

(4.19)

)0 K.

We first prove an envelope-theorem type result that implies Lipschitz continuity of the
drift of Y under uniform Lipschitz continuity of

i (!; t; x; ) in (x; ) for all i. Note

that the uniform Lipschitz condition is weaker than the assumption of uniformly bounded
derivatives of i assumed in Condition 3:1.
Lemma 4.3. If i (t; ) is uniformly Lipschitz for i = 1; : : : ; N , then Y (t; ) defined by
Y

!; t; Y; Y

=

subject to v 0 x =

max
v 0 (!; t; x; )
(x; )2RN RN d
Y and v 0

(4.20)

= Y 0;

is uniformly Lipschitz.

Proof. For simplicity of notation we will omit the (!; t) arguments. By the uniform Lip57

schitz property for i there exists a C1 2 R+ such that, for each i = 1; : : : ; N ,
i x; ~
~

i (x; )
for all (!; t) 2

n
C1 k~
x

o

xk + ~

[0; T ] , x; x 2 RN and ~ ;
~

Fix (!; t), and choose any Y; Y

~
and Y ; ~ Y

(both in R

2 RN d :
Rd ) and suppose the

maximizing arguments in (4:20) are (x; ) and x; ~ , respectively (both are in RN
~
RN d ). Denote
Y

= v 0 (x; ) ;

Y; Y

Y

= v0

~
Y ; ~Y

x; ~
~

Y
= Y 0 , v 0 ~ = ~ t 0 ). Choose i corresponding
t
~
~
to some vi > 0. Because (x; ) maximizes (4:20) for Y; Y and x; ~ for Y ; ~ Y
(and note that v 0 x =

Yt , v 0 x =
~

~
Yt , v 0

we have
Y

~
Y ; ~Y

= v0

v0

Y

Y; Y

= v 0 (x; )

x; ~
~

v0

1 n
e Y
vi i
1 n~
x + ei Y
~
vi
x+

o
~
Y ;

1 n Y0
e
vi i
o
1 n
Y ; ~ + ei ~ Y 0
vi

~Y 0

+

Y0

o

;

o

:

By Lipschitz continuity there exists a constant C2 2 R+ , independent (!; t), such that
0

B
v0 @

0

B
v0 @

n
o
1e Y Y ;
~
x+ v i
i n
o
1e
Y 0 ~Y 0
+v i
i
o
n
~
x + v ei Y Y ;
~ 1
i n
o
1 e ~Y 0
Y0
~+
vi i

1
C
A

v 0 (x; )

1
C
A

v0

C2

x; ~
~

n

C2

~
Y

n

~
Y

Y + ~Y 0

Y0

Y + ~Y 0

Y0

Combining the results yields
Y

Y; Y

Y

~
Y ; ~Y

C2
58

n

~
Y

Y + ~Y 0

Y0

o

:

o

;

o

:

We now prove Theorem 4.2.
x; ^
^

Proof of Theorem 4.2. Suppose

Y; Y

solve (4:16) and let

solve the BSDE

(5:36). The existence and uniqueness of the solution is implied by Lemma 4.2. Let Z ( N )
and c be computed as in (4:19) and (4:18), and define Z N = Y
^

v ( N )0 Z ( N ) =v N .

It is straightforward to confirm that Z; ^ so constructed solves the BSDE system

dZt =

t; M ct
^

Zt ; ^ t dt + ^ t dBt ;

~
Now consider any c 2 C and let Z; ~
~

ZT = M cT + :
^

(4.21)

denote the solution to the BSDE (4:21), with

~
~
Z; ~ ; c replacing Z; ^ ; c , and define xt = M ct Zt . Defining
~
^
~
~
~ Y 0 = v 0 ~ , and letting ~ ( N ) denote the first N 1 rows of ~ , then
t
t
t
t

~
~
Yt = v 0 Zt and
~
Y ; ~ Y solves

the BSDE

~
dYt =

8
<

8
<

0

( N) ~Y 0
v 0 @t; xt ; ~ t
~
;
:
: t

N 1
X
i=1

By (4:16) we have

v0

19
=
v i ~ i =v N A dt+ ~ Y 0 dBt ;
t
;
;

8
<

0

9
=

( N)
Y0
t; xt ; ^ t = pt + v 0 @t; xt ; ~ t
^
~
;
: t

N 1
X

vi ~ i

i=1

for some nonnegative process p, and therefore

dYt =

8
<

0

8
<

( N)
Y0
pt + v 0 @t; xt ; ~ t
~
;
:
: t

YT = v 0 :
59

N 1
X
i=1

9
=
;

1

=v N A ;

~
YT = v 0 :
(4.22)

t 2 [0; T ] ;

9
=

19
=
i ~ i =v N A dt + Y 0 dB ;
v
t
t
;
;

(4.23)

The comparison lemma of [17] applied to (4:23) and (4:22) implies Y0
N c
1 are binding) Z0 (^)

(because constraints 1; : : : ; N

~
Y0 and therefore

N c
Z0 (~). Because this holds for

all c 2 C, c must be optimal.
~
^
Constructing a solution essentially amounts to maximizing a linear combination of drift
terms. The solution xt ; ^ t , if it exists, is shown in the Lemma above to be a Lipschitz^
continuous function of Yt ; Y ; the existence and uniqueness of the BSDE solution
t
Y; Y follows from standard results.
The following example illustrates the construction of a solution.
Example 4.3. Suppose agent i’s utility process satisfies the TI specification2

i
dZt =

8
<
:

hi (t; xt )

i = 1; : : : ; N;

9
N
X j ij j0 =
1
i0
t Qt t ; dt + t dBt ;
2
j=1

i
ZT = M i ci + i ; (4.24)
T

Zt and where Qij 2 L Rd d is assumed bounded, symmetric and
h
i0
positive definite for all (w; t) and every i; j. Defining h (t; x) = h1 (t; x) ; : : : ; hN (t; x)
where xt = M ct

then (assuming the argmax is well defined)

xt = arg max
^
x2RN
has a solution of the form xt =
^

j
Qt =

N
X

i=1

ij
v i Qt ;

N
X

v i hi (t; x) ;
i=1
(t; Yt ) for some
0

QY = @
t

N
X

subject to v 0 x =

:

2
j
vj
Qt

j=1

[0; T ]

1

Yt ;

R ! RN . Defining

1 1
A ;

t 2 [0; T ] ;

2 This example violates our Lipschitz continuity condition, but existence and uniqueness
results of [4] can be used here.
60

then ^ in (4:16) is
^ i0 = v i Qi
t
t
and the BSDE (4:15) for Y; Y

v 0 h (t; (t; Yt ))

dYt =

4.4

1 Y Y
Qt t ;

i = 1; : : : ; N;

becomes
1 Y0 Y Y
Qt t
2 t

Y
dt + t 0 dBt ;

YT = v 0 :

(4.25)

Pareto optimality under linked recursive utility

In this section we use Theorem 4.1 to characterize Pareto optimal allocations with linked
recursive preferences. We fix the consumption space C = L2 RN , which Example 3.2
++
shows is extended convex, as well as the aggregate consumption process C 2 L2 R++ .
N
P i
ct = Ct : A feasii=1
i
ble allocation c is Pareto optimal if there is no feasible allocation c such that Z0 (c)
~

Definition 4.2. A consumption plan c 2 C is called feasible if
i c
Z0 (~) ; 1

i

N; with strict inequality for at least one i:

For any nonzero set of weights

2 RN we say (as in [14]) that the consumption plan
+

c is -efficient if it solves the following optimization problem:

max 0 Z0 (c) subject to
c2C

N
X

i=1

ci = C t
t

t 2 [0; T ] :

(4.26)

i
It is well known that under monotonicity and concavity of the utility functions Z0 ( ),
Pareto optimality is equivalent -efficiency. In the unlinked case, with each agent i’s aggregator a function of only i’s consumption, utility and diffusion, concavity of the aggregator and monotonicity of the aggregator in consumption imply concavity and monotonicity
i
of Z0 ( ) (see [13] for the SDU case). In the linked case, however, current comparison
61

theorems impose additional restrictions on the aggregators to obtain these properties:
n
o
2 < 1: b) (Quasi-monotonicity3 ) Fix any
Condition 4.2. a) E supt T k (t; 0; 0; 0))k
(!; t; c) 2
[0; T ] Rk ; z1 ; z2 2 RN , and 1 ; 2 2 RN d . Then, for each
k = 1; : : : ; N;
k (!; t; c; z ; )
1 1
k
k
if 4 k = k ; z1 = z2 and z1
1
2

k (!; t; c; z ; )
2 2

z2 :

Lemma 4.4. Suppose Condition 4.2 holds.
i
a) If i (!; t; ) is a concave function for all (!; t) and i then Z0 ( ) is concave for all i:
i (!; t; c; Z; ) is nondecreasing in c for all (!; t; Z; ) and i, then Z i (c) is
0

b) If

nondecreasing in c for all i.
Proof. a) Let ca ; cb 2 C and let (Za ; a ) and Zb ; b denote the solutions to the BSDEs
dZj (t) =

t; cj (t); Zj (t) ; j (t) dt + j (t) dBt;

Zj (T ) =

T; cj (T ) ;

j 2 fa; bg :
Let

~
) cb (t), Z (t) = Za (t) + (1
) b (t). Then concavity of i for all i implies

2 (0; 1) and define c(t) = ca (t) + (1
~

~ (t) =

a (t) + (1

~
dZ (t) =

n

~
t; c(t); Z (t) ; ~ (t)
~

o
pt dt + ~ (t) dBt;

~
Z (T ) =

) Zb (t),

(T; c(T ))
~

pT ;

for some nonnegative process p. Also,

dZ (t) =

f (t; c(t); Z (t) ;
~

(t))g dt +

(t)dBt;

3 The quasi-monotonicity assumption implies that
the k th row of :
4 Recall that k denotes the kth row of .
62

Z (T ) =

(T; c (T )) ;

k (!; t; c; z; :) can only depend on

Using Condition 4.2 and Condition 3.1(a), the Comparison Theorem A.1 of [50] implies
Z (0)

i
~
Z (0), which proves concavity of Z0 ( ) for each i.

b) We again use the comparison theorem of [50].
With sufficient conditions for concavity of Z i ( ), the usual separating hyperplane argument shows that Pareto optimality implies -efficiency.
i (t; !; ) is concave for all i. If c is

Proposition 4.1. Suppose Condition 4.2 and that

2 RN .
+

Pareto optimal then c is also -efficient for some
The converse of the proposition is trivial if

2 RN . However, if
++

2 RN but not
+

strictly positive, we must rely on the strong monotonicity conditions assumed for Lemma
4.4(b).
Necessary and sufficient conditions for -efficiency are obtained from Theorem 4.1.
Proposition 4.2. a) If c 2 C is -efficient then
"0
t

c1

(t; ct ; Zt ; t ) = "0 2 (t; ct ; Zt ; t ) =
t c

= "0
t

cN

(t; ct ; Zt ; t );

t 2 [0; T ] ;
(4.27)

where " is the solution of SDE (3:3) with "0 = :
j
(t)
Zi
f1; 2; : : : ; N g. If (4:27) holds and c 2 C, then c is -efficient.
b) Suppose

2 RN n 0;
+

i (t; !; ) is concave and

Proof. We apply Theorem 4.1(a) after substituting cN = C
in (4:2) is equivalent to
"0
t

n
@ (t; ct ; Zt ; t )
= "0
(t; ct ; Zt ; t )
t
ci
@ci
i 2 f1; 2; : : : ; N

1g :

63

0 for all i 6= j, i; j 2

N 1
P i
c . Then "0 c (t) = 0
t
i=1

o
(t; ct ; Zt ; t ) = 0;
cN

Corollary 4.1. Suppose for each i, i depends on ci (the agent’s own consumption) but not
on other agents’ consumption. Then Proposition 4.2 holds with condition (4:27) replaced
by
"1 11 (t; c1 ; Zt ; t ) = "2 22 (t; c2 ; Zt ; t ) =
t
t c
t
t c

= "N N (t; cN ; Zt ; t );
t
t cN

t 2 [0; T ] :
(4.28)

The following example obtains a -efficient allocation for a simple quadratic aggregator.5
Example 4.4 (quadratic aggregator). Suppose
i (t; c; Z; ) =

1
(c
2

p)0 Qi (c

p)

q i0 Z;

t < T;

i = 1; : : : ; N;

i
i 0
where Qi 2 RN N is symmetric and positive definite, q i = q1 ; : : : ; qN 2 RN and
p 2 RN . The adjoint processes satisfy the linear SDE
d"i =
t

N
X j j
"t qi dt;
j=1

"0 = ;

i = 1; : : : ; N:

(4.29)

Letting Qi denote jth row of Qi , Proposition 4.2(a) gives the FOCs
j
N
X j j
"t Q1 (ct
j=1
These N

p) =

=

N
X j j
"t QN (ct
j=1

p) :

1 inequalities together with the constraint 10 ct = Ct can be used to solve for c.

5 In this example and Example 4.6 below we assume a linear dependence of the aggregator on Z to obtain a simple expression for ". If we generalize to the additively-separable
form
i (t; c; Z; ) = 1 (c p)0 Qi (c p) + g i (t; Z; )
2
(and analogously for Example 4.6), we obtain the same expression for c, but an adjoint ",
from equation (3:3), with coefficients that depends on the solution of the BSDE (3:2).
64

P
j j
i
i
For example if Qi =diag q1 ; : : : ; qN for each i, and defining i (t) = N "t qi , then
j=1
-efficiency of c implies

ci (t)

0

1 1
1 A
pi = @ i (t)
Ct
j (t)
j=1

i
By Proposition 4.2(b), if qj

N
X

10 p

i = 1; : : : ; N:

0 for all i 6= j then we have sufficiency of the solution.

We show in the next example that the problem of -efficiency under TI preferences (and
no positivity constraint on consumption) results in either no (finite) solution, or an infinite
number of solutions.

Example 4.5 (TI preferences). We relax the requirement of strictly positive consumption
and let C = L2 RN
control to the k = N

and C 2 L2 (R). We now apply results in Section 5.5 to the
1 dimensional control c( N ) (the first N 1 elements of c). If

each agent’s aggregator depends only on his/her own consumption, then M is given by
(4:9), but we make no such assumption here. There are two cases:
a) If 0 M 6= 0 then no solution to (4:26) exists. This is seen by applying the quasilinear2
= 0 Zt c( N ) + M 0
ity property (4:8) to = M 0 to get 0 Zt c( N ) + M 0
for any c 2 C. Thus no allocation can be -efficient.
b) If 0 M = 0 then 0 Zt c( N ) +

= 0 Zt c( N ) for all

2 RN 1 , and so

if a solution exists it cannot be unique. A solution exists if there is a solution to (4:16).
This is shown by imposing (N
Ki =

1) arbitrary finite constraints on all but agent i, letting

1, and applying Theorem 4.2 to construct a solution.

The optimum, Y0 =

0 Z (^), is independent of these constraints (note that 0 M = 0 implies that v =
0 c

).

We therefore get an infinite number of solutions to the -efficiency problem, one for each
arbitrary set of constraints.
65

4.5

Optimal consumption with altruism

In this section we examine the optimal consumption and portfolio problem of an agent
when the agent’s aggregator depends on the other agents’ consumption, utility and utility
diffusion. That is, each agent is altruistic in the sense that he/she cares about the consumption, utility and risk levels of the other agents. Although the consumption processes
of the other agents are given, the consumption process chosen by agent i can impact the
utility processes of the other agents, which feeds back into agent i’s aggregator, making the
problem nonstandard.
Throughout we assume that the dimension of the consumption plan (of all agents) is N ,
but the dimension of the control of agent i is one. Each agent i trades in a complete securities market, which contains a money-market security with short-term interest rate process
r 2 L (R), and a set of d risky assets. We denote i 2 L Rd the trading plan of agent

i with i representing the vector of time-t market values of the risky asset investments.
t
Let R 2 L1 Rd represent the excess (above the riskless rate) instantaneous expected
returns process of the risky assets, and R 2 L2 Rd d the returns diffusion process,
which is assumed to be invertible for all (!; t). The planned consumption, trading and
wealth for agent i is feasible if c 2 C and the usual budget equation is satisfied:
i
i
dWt = Wt rt + i0 R
t t

R0
ci dt + i0 t dBt ;
t
t

i
ci = WT ;
T

(4.30)

as well as the integrability conditions (the latter is to rule out doubling-type strategies)
Z t

i0 R + i0 R0 R i ds < 1;
s s
s
s
02
3
n
o2
i
5 < 1:
E 4 sup
max 0; Wt
t2[0;T ]

t 2 [0; T );

We can view the wealth process (4:30) as a forward equation, starting at an initial
66

i
wealth level w0 with the terminal lump-sum balance WT consumed at T ; or we can define
agent i’s wealth process W i = W i ci as part of the pair W i ; i solving the BSDE
i0
i
i
dWt = Wt rt + t t

where t =
i=
t

R
t

i0
ci dt + t dBt ;
t

i
ci = WT ;
T

(4.31)

R0 1 R is the market price of risk, and the trading strategy financing ci is
t
t
n
o
1 i
i
. Thus Wt ci represents the time-t cost of financing ci ; s 2 [t; T ] .
s
t

We assume that is bounded, and then by Novikov’s condition there is a unique state-price
density

2 L2 (R) satisfying
d t
=
t

rt dt

0
t dBt ;

0 = 1;

(4.32)

R
i
such that Wt ci = 1 Et tT ci s ds + T ci for every ci 2 L2 (R) (see [17]). By
s
T
t
i
linearity it follows that is the gradient density of W0 ci = jci .
Agent i’s problem is to choose a consumption process ci to maximize utility subject to
the wealth constraint, taking as given c i , the consumption processes of the other agents:
i
max Z0 ci ; c i
ci :c2C

subject to

jci

i
w0 ,

(4.33)

where Z0 (c) is the initial utility specified by (3:2). The problem is nonstandard because
n
o
i (t), on Z j ; j ; j 6= i . A perof the possible dependence of agent i’s aggregator,
t t
turbation in i’s consumption plan can affect the other agent’s utility and utility-diffusion
processes, which in turn indirectly impacts agent i’s utility process.
We can adapt Theorem 4.1 to the problem as follows.

Corollary 4.2. a) (Necessity) If c 2 C solves the problem (4:33) then there is some
67

2 R+

such that6

"0 = ei ;

ci

t 2 [0; T ] ;

(t; ct ; Zt ; t )"t = t ;
o
n
i
jci
= 0;
w0

where (Z; ) solve the BSDE (3:2) and " 2 L RN solves the SDE (3:3).
j
b) (Sufficiency) Assume i (t; ) is a concave function,
(t)
0 and
Zi
for all i 6= j, i; j 2 f1; 2; : : : ; N g : If (4:2) holds then c is optimal.

j

i (t) = 0

i
If the ith aggregator i depends only on Zt and i (but can depend on the vector ct ),
t
j = 0 for j 6= i, and the FOC reduces to the standard result for generalized recursive
then "
utility (see, for example, [18] and, in the SDU case, [15] ):
i (t; c ; Z i ; i )"i =
t t t t
ci

t

d"i
0
i
i
where t = i i t; ct ; Zt ; i dt + i i t; ct ; Zt ; i dBt ;
t
t
i
Z
"t

"i = 1:
0

(4.34)

The following example applies Corollary 4.2 to a continuous-time version of [1].
Example 4.6. (Catching Up with Joneses) Letting c i =

i (t; c; Z; ) = ui (t; ci ; c i ) + q i0 Z;
i
i 0
where q i = q1 ; : : : ; qN 2 RN and ui :

X

cj , and suppose
j6=i

t < T;

[0; T ]

R

i = 1; : : : ; N;

(4.35)

R ! R. For example, the

form
ui (ci ; c i ) =

1
1

(ci )1

i

i
(c i ) ;

i

i 2 R; i 2 R
++

(4.36)

(though with q i = 0) is used in many papers including [1] and [22]. When i > 0 the
marginal utility of i’s consumption is increasing in the consumption of others, resulting in
6 Recall that e is a length-N vector with one in the ith position and zeros elsewhere.
i
68

higher consumption by i (the reverse holds for i < 0). To jointly solve for the agents’
j
optimal consumption, we define the matrix Q 2 RN N with typical element Qkj = q ,
k
and let X 2 L RN N satisfy the SDE
dXt = QXt dt;

X0 = I

(so that "t in Corollary 4.2 for agent {’s problem is the ith column of Xt ).
Corollary 4.2 we get the FOCs for optimality of ci for agent i, i = 1; : : : ; N :
(ci )
t

i c i
t
n
i wi
0

4.6

i i X ii + X
t
1
j6=i
o
i
i
jc
= 0; w0

j

j
(c )1
j t
jci ;

j

ct
i

j

j 1

0;

Applying

i = 1; : : : ; N:

ji
Xt = i t ;

Optimal portfolio with direct utility for wealth

We consider the portfolio maximization problem of a single agent in complete markets
with an aggregator that depends on current wealth.

Special cases include [49], which

examines time-additive HARA utility, but with a linear combination of consumption and
wealth replacing consumption in the aggregator; and [28], which examines (in discrete
time) an aggregator that is a function of wealth and consumption. We first present the FOCs
in the general recursive case, and consider some specializations. Example 4.7 considers
the case where consumption and wealth enter the aggregator as a linear combination, as in
[49], but with a general aggregator also dependent on utility and utility diffusion, and shows
that the solution can be obtained by solving the problem without wealth dependence after
modifying the short-rate process. Example 4.9 solves the optimal consumption/portfolio
problem for a general homothetic class of recursive utility with wealth dependence and
constrained trading.
We let N = 2 (two BSDEs) and k = 1, but for notational clarity let Z be one69

dimensional and introduce the additional BSDE (4:31) representing the agent’s wealth.
The (scalar-valued) utility satisfies the BSDE

dZt (c) =
:

where

(t; ct ; Zt ; Wt ; t ) dt + t dBt ;
R R R

[0; T ]

ZT =

T; cT ; WT :

Rd ! R. The agent’s problem is to maximize utility

subject to the budget constraint:

max Z0 (c) subject to ( jc)
c2C

w0 .

(4.37)

We solve the problem with the following corollary to Theorem 4.1:
Corollary 4.3. a) (Necessity) If c 2 C solves the problem (4:37) then there is some

2 R+

such that
2 1
c (t; ct ; Zt ; Wt ; t ) = "t ="t ;
f( jc)

w0 g = 0;

( jc)

t 2 [0; T ] ;

(4.38)

w0 ;

where (Z; ) solves the BSDE (3:2), and " = "1 ; "2 solves the SDE system7
d"1
t =
(t; ct ; Zt ; Wt ;
Z (t; ct ; Zt ; Wt ; t ) dt +
1
"t
n
o
d"2 =
"2 rt + "1 W (t; ct ; Zt ; Wt ; t ) dt "2
t
t
t
t

t ) dBt ;
0
t dBt;

b) (Sufficiency) Assume (t; ) is a concave function, and W (t)

"1 = 1;
0
"2 = :
0
0. If (4:38) holds

then c is optimal.
The first adjoint process, "1 , is the standard one for unlinked recursive utility as given
7 Note the reversal in the sign of

W , which follows because we apply Theorem 4.1 to
W to get the correct inequality constraint.
70

in (4:34) (with i = 1). The dynamics of the second adjoint process, "2 , are the same as
the state-price density

in (4:32) after adjusting the short rate for the incremental impact

of wealth on the aggregator, which is accomplished by replacing the short rate r with r +
1 2
2
W " =" (assuming " > 0). Just as with a higher interest rate, W > 0 has the effect
of deferring more consumption to the future, reducing current consumption and increasing
wealth.
Example 4.7. Suppose the aggregator depends only on a linear combination of consumption and wealth:8

(t; ct ; Zt ; Wt ; t ) =

t 2 [0; T );

(t; xt ; Zt ; t ) ;

where xt = ct + Wt , t 2 [0; T ), for some

>

(T; cT ; WT ) =

T; xT ;

1; xT = cT , and (t; ) is a concave

function. Then the FOC (necessary and sufficient) is
2 1
x (t; xt ; Zt ; t ) = "t ="t ;
Substituting W (t) =

t 2 [0; T ] :

2 1
2
x (t) = "t ="t , the dynamics of "t simplify to
d"2
t =
2
"t

(rt + ) dt

0
t dBt;

"2 = ;
0

which are the same as the dynamics of the state-price density

defined in (4:32), but with

an interest rate of r + instead of r.9 The optimal consumption problem is therefore the
same as in the case of recursive utility without wealth dependence, but with the interest
rate changed from r to r + , and the budget constraint changed from ( jc)

w0 0 to

8 Note that this specification does not fall within the TI class, which requires the time-t
aggregator to depend only on a linear combination of (ct ; Wt ; Zt ), and requires the terminal
utility be a linear in cT .
9 That is, "2 = e t
t , t 2 [0; T ]. An alternative approach to the problem is to use
t
an isomorphism as in [42].
71

"2 jx

w 0 "2 .
0

Example 4.8. Suppose the aggregator takes the form

(t; ct ; Zt ; t ) = f (t; Zt ; t ) +
where

(ct )1

+ (Wt )1
1

;

> 0 controls the curvature of utility over both consumption and wealth,

> 0 is

a scaling parameter, which allows us to control the “intensity” of the agents’ direct wealth
preference. Corollary 4.3 implies that optimal consumption satisfies
1=
ct = "1 ="2
:
t t
The final example uses Theorem 4.2 in Section 5.5 to solve the optimal consumption
and portfolio problem with homothetic preferences.
Example 4.9 (Homothetic wealth-dependent utility). Suppose the homothetic specification
dUt
=
Ut
where g :

c W
g t; t ; t
Ut Ut
[0; T ]

U
+ q t; t

R2
++ ! R, q :

U
dt + t 0 dBt ;
[0; T ]

UT = CT ;

(4.39)

Rd ! R, g (t; ) and q (t; ) are

concave, and g satisfies the Inada conditions in the ct =Ut argument.

The special case

with no dependence on Wt =Ut is examined in [43]. Epstein-Zin utility corresponds to the
power form of g in (4:42) below with

U
= 0 (no dependence on Wt =Ut ) and q t; t

=

U0 U
t t for some > 0. We also relax the assumption of complete markets.
i
Defining the investment proportion process by t = i =Wt , and the consumptiont
to-wealth ratio t = ct =Wt , the budget equation (4:30) can be written
dWt
= rt
Wt

0 R
0 R0
t + t t dt + t t dBt ;

WT = CT :

(4.40)

Because the utility and wealth aggregators fall within the TI class (after transforming
72

W , U , and c to logs),10 the problem can be solved using the dynamic programming approach of Section 5:5. We impose possible trading restrictions by assuming that t 2 K,
t 2 [0; T ], for some convex set K. The homothetic form implies Ut = t Wt for some
satisfying
d t
= t dt + t 0 dBt ;
t

T = 1:

The optimality condition (from Theorem 4:2) is

max
t = x>0; y2K rt

x + y0

R + R0
t t
t

+ g t;

x
t

;

1
t

R
+ q t; t + t y

;

(4.41)
with the optimal ( t ; t ) representing the maximizing arguments. The additive separability
of the aggregator implies that we can separately solve for t and t :
x 1
^t = arg max g t; ;
x ;
x>0
t t
n
o
R
^ = arg max y 0 R + R0
+ q t; t + t y :
t
t
t t
y2K
Given these solutions, we obtain t as a function of t , and then solve for the BSDE for
;

to complete the solution.
If g has the power form

g (t; z1 ; z2 ) =

8
>
>
>
<

>
>
>
: t

t1

1

1
1+ ln

(

z1 z2

ct
Ut

1
1+

+ 1+ ln

)

1

Wt
Ut

if 1 6=
if

> 0;
(4.42)
= 1;

10 Let Z = [ln (U ) ; ln (W )]0 and apply a logarithmic transformation to consumption.
t
t
t
Then M = (1; 1)0 , v = (1; 1)0 and Yt = v 0 Zt =ln( t ). As stated in Section 4:18, the
dynamic programming approach in Theorem 4.2 extends easily to additional controls, such
as the constrainted portfolio choice we introduce here.
73

with

+

> 0, we get
^t =

t
1+

1
t

1+
+

:

In the log case ( = 1) this simplifies to ^t = t = (1 + ), which is invariant to (and
therefore invariant to the dynamics of R , r and R ), and decreasing in , reflecting a
desire to postpone consumption and increase wealth when more weight is placed on wealth
in the aggregator.

74

5
Continuous Time Principal-Agent Problem

5.1

Introduction

We study the principal-agent problem with moral hazard in a continuous-time Brownian
filtration with recursive preferences on the part of both principal and agent, and pay over
the lifetime of the contract. Previous work has considered only additive utility, which, as
is well known, arbitrarily links intertemporal substitution and risk aversion (see, for example, [19]). Yet time-additivity offers essentially no advantage in tractability because agent
optimality induces recursivity to the principal’s preferences even in the additive case. We
allow both principal and agent preferences to be within the generalized recursive utility
75

class, which was introduced in [34] to unify the stochastic differential utility (SDU) formulation of [13], and the multiple-prior formulation of [5]. Unlike the additive class, the
recursive class allows distinct treatment of aversion to variability in consumption across
states and across time.

Furthermore, the class can accommodate source-dependent (do-

mestic versus foreign, for example) risk aversion, differences in agent and principal beliefs,
as well as first-order risk aversion (which imposes a higher penalty for small levels of risk)
in addition to the standard second-order risk aversion.1

Also, [47] shows that SDU, a

special case of the recursive class, includes the robust control formulations of [2]; [24]; and
[37].
In the principal-agent problem with moral hazard, a utility maximizing principal pays
a compensation process to an agent in order to induce effort (which typically increases
expected future cash flows). However the principal faces two constraints. First, because
effort is assumed noncontractible, the contract must satisfy an incentive compatibility condition that the agent, faced with a particular compensation process, will choose effort that
maximizes his/her own utility. Second, because the agent has employment opportunities
elsewhere, the agent’s initial utility must exceed some fixed amount. In the continuoustime Brownian version, first examined in [27], the impact of effort choice is typically modeled as an equivalent change of measure (that is, the agent’s efforts change the probabilities
of the states), which changes the drift of the driving Brownian processes. This is a convenient way to model, for example, the impact of effort on the growth rate of a cash flow
process.
We derive necessary and sufficient conditions for both agent and principal optimality,
and show that the first-order conditions (FOCs) for the principal’s problem take the form
of a forward-backward stochastic differential equation (FBSDE). The utility processes are
backward systems, and these are coupled together with a forward equation which incor1 See [44] and [48]. We consider only second-order risk aversion in this paper, but
extensions to the first-order case (modeled by kinks in the aggregator) can be handled
along the lines of [44].
76

porates both the impact of agent effort on principal utility and the agent’s participation
constraint. We also provide a dynamic programming proof of sufficiency.
When agent and principal preferences are translation invariant (TI), a class of preferences introduced in [44] as an extension of time-additive exponential utility, the system
uncouples and dramatically simplifies to the solution of a single backward stochastic differential equation (BSDE). This BSDE can be interpreted as a subjective cash-flow presentvalue process, and incorporates a mixture of agent and principal preferences. Construction
of a solution in the TI case is straightforward and the required technical conditions are less
stringent. We illustrate with a number of examples with quadratic risk-aversion and effort penalties, and obtain closed-form solutions for some parametric examples, including
Ornstein-Uhlenbeck and square-root cash-flow dynamics. In the quadratic class we obtain
a simple sharing rule for the volatility the subjective present-value process. These sharing
rules depend only on risk aversion and effort penalty processes, not on preferences for intertemporal substitution. In only very special cases do these volatility sharing rules imply
linear sharing rules for the cash flows themselves.
As in [41], we consider a general (non-Markovian) Brownian setting, and derive firstorder conditions for the agent and principal’s problems. [41] use the martingale approach
for stochastic control theory to solve for the first-order condition of optimality under exponential utility and terminal consumption only. Our paper considers generalized recursive
preferences and lifetime and terminal consumption. Methodologically, we rely on a combination of the utility gradient approach and dynamic programming.
Many other papers have considered variations of the continuous-time principal-agent
problem, but have been focused on particular applications in a Markovian setting. See, for
example, [40] (the agent controls the drift of an output process with constant volatility),
[11] (extends [10] to continuous setting), and [25] and [26] (the agent controls the drift of
the firm’s cash flow process with a binary effort choice).
[9] consider a general Brownian setting, but with agent and principal maximizing ex77

pected utility of lump-sum consumption. They provide a necessary condition for optimality in the general case as a solution of system of coupled FBSDEs and a maximal principle;
these are also sufficient under regularity conditions. They obtain an essentially closed-form
solution in the case of quadratic effort penalty (see Example 5.4 below).

5.2

Setup and Statement of the Problem

We will use the same setting as Chapter 3. The definition of utility from Chapter 3 will
be assumed in this chapter. The agents and the principal’s preferences will be assumed to
follow generalized recursive utility.
For any subset S of Euclidean space, let L (S) denote the set of S-valued processes,
and, for any p

1,

Lp (S) =

(

x 2 L (S) : E

n

"Z
T

Lp (S) = x 2 Lp (S) : E

0

#

)

kxt kp dt < 1 ;

o
xT p < 1 ;

where kxt k denotes Euclidean norm. Note that L2 (R) is a Hilbert space with the inner
product
(x jy ) = E

"Z

#

T

0

xt yt dt + xT yT ;

x; y 2 L2 (R) :

Finally, define

E =

8
<
:

2

0

x 2 L (R) : E 4exp @

13

sup jxt jA5 < 1 for all
t2[0;T ]

e
We re-define the set of consumption plans as the set C

9
=
>0 :
;

L2 (C) where C

R (in

e
typical applications, either C = R or C = R++ ). For any c 2 C, we interpret ct as a
e
consumption rate for t < T , and cT as lump-sum terminal consumption. Let C denote
78

the set of intermediate consumption plans (i.e. ct , 0
e
plans as E = fe 2 L2 (E)g for some convex set E

t < T ). We define the set of effort
Rd (there is no lump-sum terminal

effort).

The impact of agent effort is modeled as a change of probability measure. Let
Z t

e
Zt = exp

e0 dBs
s

0

!
Z
1 t
kes k2 ds :
2 0

e
e
We assume throughout that Z e is a martingale (equivalently, EZT = 1) for all e 2 E. One
of the well known condition for Z e to be a martingale is that e satisfies Novikov’s condition.

Define the probability measure P e (with expectation operator E e ) corresponding to effort
e by
dP e
e
= ZT :
dP
e
Girsanov’s Theorem implies Bt = Bt

Zt

es ds is standard Brownian motion under P e

0
with respect to the filtration fFt : t 2 [0; T ]g.
Preferences are assumed to be in the generalized recursive utility class.

Given the

e
e
consumption stream c 2 C paid by the principal, and effort level e 2 E chosen by the agent,
the agent’s utility U (c; e) is the first element of the pair U; U

assumed to uniquely

satisfy the BSDE

dUt =
The function F :
P

e
F t; ct ; et ; Ut ; U dt + U 0 dBt ;
t
t
[0; T ]

C

E

UT = F T; cT :

(5.1)

R1+d ! R is called the aggregator and is

B(R2+2d ) measurable, where P is the predictable -field on

lump-sum terminal consumption depends on only

[0; T ]. Utility of

and cT (there is no lump-sum terminal

effort).
Agent effort is not contractible, but can be influenced by the principal through the con79

e
sumption process paid to the agent. For any agent effort choice e 2 E and consumption

e
stream c 2 C paid to the agent, the principal’s utility V (c; e) is the first element of the pair
V; V

assumed to uniquely satisfy the BSDE

dVt =
where G :

[0; T ]

G t; ct ; Vt ; V
t
C

e
dt + V 0 dBt ;
t

VT = G(T; cT );

(5.2)

R1+d ! R is the principal’s aggregator and is P

measurable. Terminal utility again depends on only

B(R2+d )

and cT .

We will assume throughout this paper that F and G are concave and differentiable in
c; e; U; V; : Also Fc > 0 and Gc < 0 (the agent’s consumption is paid by the principal).
Example 5.1 (Time-Additive Utility). The case of time-additive preferences with quadratic
penalty for agent effort corresponds to
U
t U

F (t; c; e; U; ) = ut (c)
F T; cT = uT cT ;

1 0
qe e;
2

G (t; c; V; ) = vt (Xt

G T; cT = vT XT

c)

V
t V;

t<T

cT ;

for some increasing and concave functions vt ( ) and ut ( ), some q 2 R++ , discount
processes U and V , and cash-flow-rate process X satisfying
dXt = X dt + X0 dBt ;
t
t
with X 2 L1 (R) and

X 2 L
2

Rd .

With sufficient integrability, agent utility

satisfies
e
Ut = Et

(Z
T
t

Rs U
e t u du us (cs )

1 0
qe es ds + e
2 s

)
RT U
du u c
t u
;
s T

with an analogous expression (with q = 0) holding for principal utility. The principal
consumes at the rate Xt

ct , for t < T , and the lump-sum amount XT
80

cT , which is

what remains of X after paying the agent. Agent effort e changes the probability measure
to P e , resulting in a change in the drift of the cash-flow process of e0 X .
t t
We next define the set of feasible consumption and effort plans. We will use the theory
on BSDEs with aggregators which have quadratic growth in volatility. This theory was
developed in [32] for the case of uniformly bounded terminal values and later extended in
[4] to the case of terminal values with exponential moments. Their results show that our
definition of feasibility given below implies existence and uniqueness of solutions to the
BSDEs (5:1) and (5:2) (as well as allow the application later of a comparison theorem for
BSDEs). We note that stricter conditions on the aggregators (such as Lipschitz continuity) are associated with weaker integrability conditions on UT and VT for existence and
uniqueness to hold (see [17]). Furthermore, as less restrictive conditions for existence and
uniqueness of BSDE solutions are developed (hopefully), the restrictions in the following
definition of a feasible plan can be relaxed, and our results will hold with similar proofs.
e
Definition 5.1. (c; e) 2 C

e
E will be called a feasible consumption and effort plan with

respect to the aggregators F and G if there exist a process (t)
;

0 and two constants

> 0 such that following hold:
a) for all (t; u; v; u0 ; v 0 ; ) 2 [0; T ]
F (t; ct ; et ; u; )
(u

u0 + v

Rd+4 ;

F (t; ct ; et ; u0 ; ) + G(t; ct ; v; )

G(t; ct ; v 0 ; )

v 0 );

b) for all (t; u; v; ) 2 [0; T ]

Rd+2 ;

jF (t; ct ; et ; u; )j + jG(t; ct ; v; )j

(t) + (juj + jvj) +

2

j j2 ;

RT
c) 0 (s)ds , UT , and VT have exponential moments of all order.
81

n
o
e
e
Define C = c 2 C j 9e 2 E s.t. (c; e) is feasible and
n
o
e
e
E = e 2 E j 9c 2 C s.t. (c; e) is feasible : For simplicity of presentation we will as-

sume the product structure, i.e.
n
o
e E j (c; e) are feasible , but the proofs all go through without this
e
C E = (c; e) 2 C
assumption.

Remark 5.1. Let A be progressively measurable subset of
(c1 ; e1 ); (c2 ; e2 ) 2 C

[0; T ]: Given two plans

E; we can merge them into one plan defined as: (c; e) = (c1 ; e1 )

1A + (c2 ; e2 ) 1Ac : It is easy to see that (c; e) 2 C

E as well.

Remark 5.2. As was mentioned before it follows from [4] that for any (c; e) 2 C
there is a unique solution U; U
E

L2

Rd to the BSDE (5:2).

2E

L2

Rd to the BSDE (5:1) and V; V

E
2

Principal optimality, in Theorem 5.2 below, is obtained using an extension of the KuhnTucker Theorem (see Chapter 3), which relies on the assumption that C is an "extended
convex" set, which we define as follows.
Definition 5.2. X, a collection of stochastic processes, is extended convex if for all x1 ; x2 2
X there is a process = (!; t; x1; x2 ) > 0 such that
)x2 2 X

x1 + (1
for each process

= (!; t) that satisfies

1+ :

Given any c 2 C, the agent chooses effort to maximize utility:
U0 (c) = sup U0 (c; e) :
e2E
Let
e (c) = fe 2 E jU0 (c) = U0 (c; e)g
82

(5.3)

denote the set of optimal agent effort processes induced by the consumption process c.
The principal’s problem is to choose the optimal consumption of the agent subject to the
participation constraint that the initial agent utility must be at least K:

sup
V0 (c; e) subject to U0 (c)
c2C;e2e(c)

5.3

K:

Agent Optimality

This section derives a necessary and sufficient condition for agent optimality given any
consumption plan offered by the principal. We show that optimality is essentially equivalent to choosing effort to minimize the instantaneous drift of U (that is, maximizing
F t; ct ; et ; Ut ; U
t

+

U 0 e at each (!; t)). A necessary and sufficient characterizat t

tion of agent optimality is given in the following theorem.
Theorem 5.1 (Agent Optimality). Fix some c 2 C . Then e 2 E is optimal if and only if
F ct ; et ; Ut ; U + U 0 et
t
t

F ct ; et ; Ut ; U + U 0 et ,
~
t
t ~

t 2 [0; T );

for all e 2 E,
~
(5.4)

where Ut = Ut (c; e) and U = U (c; e) solve the BSDE (5:1).
t
t
Proof. See the appendix.

If the optimal effort is interior (that is et 2int(E), t 2 [0; T )), then (5:4) is equivalent
to
Fe t; ct ; et ; Ut ; U
t

= U:
t

(5.5)

At each (!; t), the agent chooses the effort level that, at the margin, equates the instantaneous cost,

Fe , and the instantaneous measure-change benefit,

U , per additional unit
t

effort. That is, optimal effort equates incremental cost to the sensitivity of the agent’s utility
83

to unit Brownian shocks. The policy is particularly simple because it depends only on the
current time, state and values of ct ; Ut ; U .
t
Example 5.2. Suppose E = Rd and the aggregator is separable in effort with a quadratic
penalty:
F !; t; c; e; U; U

= h !; t; c; U; U

1 0 e
e Q (!; t) e;
2

where Qe 2 L Rd d is positive definite. Optimal agent effort is then linear in U :
et = Qe
t

1 U:
t

If F is a deterministic function (does not depend on !), then all uncertainty and therefore
all effort is driven by the consumption process. For example, if F is deterministic and c satisfies the Markovian SDE dct = c dt + c0 dBt for some c 2 R and c 2 Rd , then agent
utility under optimal effort will take the form Ut = g (t; ct ) for some deterministic function
g increasing in consumption, and optimal effort will satisfy et = gc (t; ct ) Qe
t

1 c.
t

Our framework is sufficiently flexible such that factors other than consumption can
induce effort. For example, reputation concerns or some stochastic cash flow process not
provided by the principal can be incorporated.
The following example considers a quadratic risk-aversion penalty combined with a
quadratic effort penalty.
Example 5.3. Suppose we further specialize Example 5:2 by assuming a quadratic utility
penalty:
F !; t; c; e; U; U

= h (!; t; c; U )

1 0 e
e Q (!; t) e+p (!; t)0 U
2

1 U0 U
Q (!; t) U ;
2

where Qe ; QU 2 L Rd d are positive definite and p 2 L Rd . The process QU
represents aversion towards the d-dimensions of risk, while p represents beliefs different
84

from the true probability measure P (the agent views P p as the true probability measure).
Then under the optimal effort process et = Qe
t
dUt =

h (!; t; c; U ) + pt U
t

1 U , agent utility satisfies
t

1 U0
QU
t
2 t

Qe
t

1

U
t

dt + U 0 dBt ;
t

UT = F T; cT ;
which is the same as the zero-effort utility function except with a modified risk-aversion
coefficient.
On one hand, the quadratic effort penalty combines with the risk aversion penalty to
increase the quadratic penalty on U . On the other hand, an increase in U also increases
t
t
the private benefit to agent effort (in typical applications, this benefit is through the share
of some cash flow process promised to the agent, and the impact of effort on the cash
flow drift), which, together with the higher level of effort induced, results in a positive
contribution to utility, also quadratic in
Qe
t

U.
t

If Qe is small enough (that is, if QU
t

1 is negative definite), then effort choice can induce a preference for risk.

The dependence of agent effort on the volatility term

U , combined with the depen-

dence of principal utility on effort (via the change of measure), causes a dependence of the
principal’s utility on the agent’s volatility, and induces a (bivariate) recursiveness to utility
under optimal agent effort even if none existed for a given effort process. We will see more
about this later in the paper.

5.4

Principal Optimality

The principal’s problem, of choosing the consumption/compensation process c to maximize the principal’s utility given the agent’s optimal effort, is far more complicated than
the agent’s problem of choosing effort taking the consumption process as given. The principal must balance the reduction in his own utility from paying an extra dollar against the
85

benefits of increased agent effort as well as the slackened participation constraint resulting
from the increased utility of the agent. We express the solution as an FBSDE, which links
the BSDEs of both principal and agent utilities together with a forward equation representing the sensitivity of principal’s utility to a unit change in the agent’s utility. Optimal
consumption at each (!; t) depends not only on the utility and utility diffusions of both
principal and agent, but also on the value of this Lagrange multiplier process, which incorporates the shadow price of the participation constraint as well as the dependence of the
principal’s utility on agent effort.
We first obtain necessary and sufficient conditions for optimality using the utility gradient approach which originated in the portfolio optimization problem in [8] and [30] for
the case of additive-utility, and extended for non-additive utilities in [46]; [15]; [18] (which
allows non-linear wealth dynamics); [43]; [44] (the latter two allow constrained markets);
and [45] (which considers constraints and a jump-diffusion setting). The gradient approach
applied in the portfolio setting essentially amounts to choosing the consumption stream to
match marginal utility (the utility gradient density) to marginal price (the state-price density). In the principal-agent setting, the agent utility process plays a role analogous to the
that of the wealth process, and the optimality conditions take a similar form. In addition
to utility gradient approach to be developed in Section 5.4.1, we will present a dynamic
programming approach in Section 5.4.2, from which we obtain sufficient conditions for
principal optimality under different regularity conditions.

5.4.1

Utility gradient density approach

We begin by substituting the agent’s optimal effort into the utility BSDEs. Define I :
[0; T ]

C

R1+d ! E by

I !; t; c; U; U

n
o
= arg max F !; t; c; e; U; U + U 0 e ;
e2E
86

t 2 [0; T ):

(5.6)

By Theorem 5.1, if I is well-defined (the arg max above exists and is unique) then et =
I !; t; c; U; U is the optimal choice of the agent effort. That is why we assume
throughout the following condition.2

C Rd+1 ; I (!; t; c; u; ) is well
n
o
U ; t 2 [0; T ] 2 E;
defined and I 2interior(E). Also for each c 2 C , I t; ct ; Ut ; t

Condition 5.1. For each (!; t; c; u; ) 2
where U; U

[0; T ]

is solution of (5:1) with et replaced by I t; ct ; Ut ; U (the solution to
t

equation (5:8) below).

It follows from Condition 5.1 that e(c), the set of optimal effort processes induced by c
(defined in (5:3)), contains exactly one process.
Sufficient conditions for a unique interior (with respect to E) solution are that F is
strictly concave in e and that Fe !; t; c; ; U; U maps E onto Rd for each (!; t; c; U; ).
Substituting optimal effort into the BSDEs (5:1) and (5:2), the principal’s problem is

sup V0 (c) subject to U0 (c)
c2C
where U; U ; V; V

K

(5.7)

satisfy the BSDE system

dUt =

F t; ct ; Ut ; U dt + U 0 dBt;
t
t

dVt =

G t; ct ; Vt ; V ; Ut ; U dt + V 0 dBt ;
t
t
t

UT = F T; cT ;

(5.8)

VT = G T; cT ;

and where we have used the abbreviations Vt = Vt (c), Ut = Ut (c),

V =
t

V (c) and
t

2 Extensions to corner solutions in (5:6) as well as nondifferentiable aggregators are
straightforward. The utility gradient approach can be handled along the lines of [43],
replacing derivatives with appropriately defined superdifferentials. The dynamic programming approach below doesn’t use the differentiability assumption, and constraints are simple to impose.
87

U = U (c), and the modified aggregators
t
t
F !; t; c; u; 1 = F !; t; c; I !; t; c; u; 1 ; u; 1 + 10 I !; t; c; u; 1 ;
G !; t; c; v; 2 ; u; 1 = G !; t; c; v; 2 + 20 I !; t; c; u; 1 :

Condition 5.1 implies a unique solution of (5:8) for each c 2 C.
The agency problem induces a dependence of the principal’s aggregator on the utility
and utility-diffusion term of the agent.

This follows because agent effort, which is a

function of both the agent’s utility and utility-diffusion term, affects the Brownian motion
drift, which impacts the principal’s utility.
The principal’s optimality conditions below are expressed in terms of the gradient density (or Gateaux derivative) or supergradient density of a linear combination of agent and
principal utilities. The general definitions of these densities follow.

Definition 5.3. Let

: C !R be a functional. For any c 2 C, the process

supergradient density of

at c if

(c + h)

and

2 L2 (R) is a

(c)

( jh)

for all h such that c + h 2 C;

2 L2 (R) is a gradient density at c if
( jh) = lim
#0

(c + h)

(c)

for all h such that c + h 2 C for some

> 0.

The computation of the gradient and supergradient densities requires the following R2 0
valued adjoint process "t = "V ; "U , with some initial value "0 2 R2 and dynamics
t t
0

1

0

1
0 dB
0
0
t
B GV (t)
C
B G V (t)
C
d"t = @
A "t dt + @
A "t ;
GU (t) FU (t)
G U (t)0 dBt F U (t)0 dBt
88

(5.9)

where GV (t) = @G (t; ct ; U; U ; V; V ) and similar definitions for rest of the terms in
@V
(5:9) :
The following condition imposes sufficient smoothness and integrability on the aggregators for existence of gradient densities and supergradient densities.
Condition 5.2. Define

(t) = (F (t); G(t)), and let " satisfy the SDE (5:9) with initial

value "0 . The conditions below hold 8c 2 C:
a) Gradient density conditions:
For every t 2 [0; T ],

(t; ) has uniformly bounded continuous partial derivatives

with respect to (c; U; V; U ; V ): (The bounds do not depend on !.)
f (t; ct ; 0; 0; 0; 0) ; t 2 [0; T )g 2 L2 (R2 ):
E[
(T; cT ) 2 ] < 1:
(The last two conditions follow from Definition 5.1 and are brought here for convenience of presentation.)
b) Supergradient density conditions:
n

i (t; ) is a concave function in (c; U; V; U ; V ); for i 2 f1; 2g.
o
o
n
i (t) ; t 2 [0; T ] is locally bounded3 for i 2 f1; 2g; Z 2 U; V; U ; V ;
Z

where i (t) is calculated in U (c); U (c); V (c);
Z
o
n
ht i (t); t 2 [0; T ] 2 L2 (R) ; E supt2[0;T ]
c

V (c) , a solution of (5:8) :
2
"i
< 1 for all h such that
t

c + h 2 C; i 2 f1; 2g.

It is convenient to express the FOCs in terms of the gradient of a linear combination
of utilities (though, as remarked below, it could instead be stated in terms of the individual
gradients of the agent and principal utilities).
Lemma 5.1 (Gradient and supergradient densities). Suppose c 2 C, U; U ; V; V

sat-

isfies the BSDE system (5:8), and " satisfies (5:9) with initial value "0 2 R2 .
+
3 A process X(t; !) 0

t T; is called locally bounded if 9 k " T; k are stopping
times such that X(t ^ k ; !) is bounded 8k:
89

Gc (t) ; Fc (t) "t ; t 2 [0; T ] is a gradient density
of [V0 (c) ; U0 (c)] "0 at c for any "0 2 R2 :
+
a) If Condition 5.2 (a) holds, then

Gc (t) ; Fc (t) "t ; t 2 [0; T ] is a supergradient
density of [V0 (c) ; U0 (c)] "0 at c for any "0 2 R2 :
+
b) If Condition 5.2 (b) holds, then

Proof. See Lemma 3.1 and 3.3 in chapter 3.
The main result of the section, providing the first-order conditions (FOCs) for principal
optimality, follows.
Theorem 5.2. Let c 2 C, U; U ; V; V

solve the BSDE system (5:8), and suppose "

solves the SDE (5:9).
a) (Necessity) Suppose Condition 5.2 (a). If c solves the principal’s problem (5:7) then
there is some

2 R+ such that
"0 = (1; )0 ;

t 2 [0; T ] ;

Gc (t) ; Fc (t) "t = 0;

fU0 (c)

Kg = 0;

U0 (c)

(5.10)

K:

b) (Sufficiency) Suppose Condition 5.2 (b). If (5:10) holds then c is optimal.
Proof. Follows from theorem 4.1 in chapter 4.
Remark 5.3. The FOCs (5:10) can be restated as follows.

If U ; V 2 L2 (R) are

gradient densities or supergradient densities of U and V , respectively, at c, then (5:10) is
equivalent to
V =

U;

fU0 (c)

Kg = 0:

(5.11)

V
U
The equivalence follows because for i 2 f1; 2g, t = Gc (t) ; Fc (t) "1 , and t =
t
Gc (t) ; Fc (t) "2 and "i satisfies (5:9) with initial values "1 = (1; 0)0 and "2 = (0; 1)0 .
t
0
0
1 + "2 where " = (1; )0 .
The linearity of the adjoint processes (5:9) implies that "t = "t
0
t

90

The FOCs are the natural extension of the static problem, with

representing the

shadow price of the participation constraint.
Concavity of

(t) = (F (t); G(t)), which is assumed in Condition 5.2 (b) for the suf-

ficiency part of the Theorem, is violated in many applications. A dynamic programming
derivation of sufficiency based on weaker conditions (see Remark 5.5) is introduced in
Section 5.4.2 below.
The dimensionality of the solution can be reduced by expressing the FOCs (5:10) in
terms of a ratio. Define the process4

t=

Gc t; ct ; Vt ; V ; Ut ; U
t
t
;
Fc t; ct ; Ut ; U
t

t < T;

T =

Gc T; cT
Fc T; cT

=

Gc T; cT
;
Fc T; cT
(5.12)

which under the FOCs (5:10) satisfies
"U
t
t= V;
"t
where "0 = (1; )0 for some

(5.13)

0. The dynamic programming argument in Section

5.4.2 justifies the interpretation of t as the Lagrangian of the time-t principal optimization
problem, representing the rate of change in Vt per unit reduction in Ut . At the margin it is
given by the ratio of the sensitivities of the corresponding utilities to unit changes in time-t
consumption, which, in turn, equals the ratio of the marginal aggregators. Ito’s lemma
implies the following dynamics of :

d t=

n

o
GV (t) + GU (t) G0 V (t) t dt + t 0 dBt ;
t
n
o
where t = t F U (t) G V (t) + G U (t) :

t FU (t)

(5.14)

We will assume that (5:12) can be inverted to solve consumption as a function of
4 Note that F = F and recall the assumption F > 0.
c
c
c
91

V
U
t ; Vt ; t ; Ut ; t :
Condition 5.3. There exists
satisfies (5:12) with ct =

[0; T ] R2d+3 ! R so that (t; ct ; t ; Vt ; V ; Ut ; U )
t
t
!; t; ; V; V ; U; U .
:

Given Condition 5:3 we can express the first-order conditions (5:10) as an FBSDE
system for U; U ; V; V ;

dUt =

:

F t; ct ; Ut ; U dt + U 0 dBt;
t
t

UT = F T; cT ;

G t; ct ; Vt ; V ; Ut ; U dt + V 0 dBt ; VT = G
t
t
t
8
>
<
t FU (t)
t GV (t) + GU (t)
d t=
> G0 (t)
:
t F U (t)
t G V (t) + G U (t)
V
0
+ t F U (t)
t G V (t) + G U (t) dBt ;
0=
dVt =

U0

K;

ct =

(U0

(5.15)
T; cT ;
9
>
=
dt
>
;
0;

K) = 0;

t; t ; Vt ; V ; Ut ; U ;
t
t

cT =

t; T :

The solution is complicated because the backward utility equations and the forward
equation for

are coupled, requiring the system to be simultaneously solved.

Remark 5.4. (a) The optimality conditions can be expressed in terms of the original aggregators F and G as follows. The equation

Fc (t) t = Gc (t) is equivalent to (substituting

U)

Fe (t) =

0 = Fc (t) t + Gc (t) + V 0 Ic (t) :
t

(5.16)

Furthermore, G U (t) = I (t)0 V implies
d t=

n

t FU (t)

V0
t GV (t) + t IU (t)

where t = t F (t)
92

G (t) + I (t) 0 t

G (t) + I (t)0 V :
t

o

dt + t 0 dBt ;
(5.17)

(b) A sufficient condition for the participation constraint to bind is easily obtained from
(5:16). If

V 0 I (t)
t c

0 for all t (for example, if Ic = 0), then (5:16) implies

> 0

(recall Fc > 0 and Gc < 0). In particular, 0 > 0 which implies a binding constraint.
The first-order condition for optimal consumption (5:16) equates the instantaneous net
benefit per unit change of time-t pay to zero.

The term Gc (t) +

V 0 I (t) is the dit c

rect impact on principal utility per unit extra consumption for the agent, the second term
representing the direct effect of the incremental pay on agent effort and its impact on the
principal’s utility (via the change of measure). In typical applications, however, Ic = 0 and
the incentive effect of compensation is through the agent’s volatility, U (see Example 5:3
= fQe (!; t)g 1 U ). Increasing agent pay by promising,
t

above in which I !; t; U
t

say, a larger share of the cash flows increases the (absolute) sensitivity of agent volatility;
this induces additional effort, which increases the drift of the cash flow process and increases both agent and principal utility. At the optimum, this effect is captured by the term
Fc (t) t , with the Lagrangian t representing the benefit to the principal from increasing
agent utility, which incorporates incentive effects of extra pay as well as the shadow price
of the participation constraint.

5.4.2

Dynamic Programming Approach

Our first goal is to reformulate the agent’s utility process as a forward equation. Recall
that for any consumption plan c 2 C paid by the principal there is a unique agent utility

and volatility pair U; U

2E

L2

Rd solving the BSDE (5:1) with, by Theorem

5.1 and Condition 5.1, optimal agent effort et = I t; ct ; Ut ; U . In Remark 5.4(b) we
t
presented a sufficient condition for

> 0, which implies a binding participation constraint.

The principal’s problem is made amenable to the dynamic programming approach by reformulating the agent’s utility equation as a forward SDE (starting at K), and changing
the principal’s controls from lifetime consumption to intermediate consumption and agent
93

utility volatility. That is, we proceed as if the principal can control the agent’s utilityvolatility process U in addition to intermediate consumption. The terminal value of the
agent’s utility gives the unique cT that corresponds to these controls, which, together with
intermediate consumption, forms the pay package offered to the agent. This approach is in
the spirit of [41] and [27]. The problem is analogous to the optimal portfolio/consumption
problem, with the forward equation U playing the role of a non-linear wealth equation, K
representing initial wealth, and U the portfolio vector.
e
We start with the primitive intermediate consumption space C

(recall that terminal

e
consumption is not included in c 2 C ) and the space of agent utility-volatility processes
as L2

Rd . A principal’s plan will be some pair c; U

any such plan agent utility solves the forward SDE

dUt =

e
F t; ct ; et ; Ut ; U dt + U 0 dBt; ;
t
t

L2

Rd , and for

U0 = K;

(5.18)

e
2C

where optimal agent effort is et = I t; ct ; Ut ; U . The lump-sum terminal consumption
t
corresponding to the plan is cT = F 1 T; UT (in the language of the analogous portfolio
problem, cT is "financed" by the diffusion process c; U ).
We would like the utility gradient density approach which we developed in Section
5.4.1 to be consistent with the dynamic programming approach. However using (5:18)
with any

U 2L
2

e
Rd will give cT = F 1 T; UT , which combined with c 2 C

doesn’t necessarily belong to C (i.e. the solution is not feasible). The following definition
guarantees that the solution we get from current approach is feasible.

Definition 5.4.
(c; e) 2 C

c; U

e
2 C

L2

Rd

will be called a viable principal’s plan if

E, where cT = F 1 T; UT and et = I t; ct ; Ut ; U , with Ut satisfying
t

equation (5:18). We will denote the class of viable principal’s plans by A:
94

The principal’s problem in the current setting is
V0 c; U :
sup
c; U 2A
The restriction in (5:19) to c; U

(5.19)

2 A ensures that in the case of a binding participation

constraint, the solution we get from the two approaches (gradient density and dynamic
programming) will be identical.

The solution of (5:19) is implemented as follows: The principal specifies the intermediate consumption plan and the agent’s volatility. Then the agent maximizes his/her utility
by choosing the optimal effort et = I t; ct ; Ut ; U . The principal computes the tert
minal consumption consistent with c; U and a binding participation constraint using
equation (5:18), yielding cT = F 1 T; UT : At this stage the principal’s utility, V (t);
solves equation (5:2) with et = I t; ct ; Ut ; U .
t
Having reformulated the problem, we will use dynamic programming to solve the principal’s problem. We start by defining

Yt = Vt + t (Ut
where

K) ;

t 2 [0; T ] ;

satisfies the BSDE
d t = t dt + t 0 dBt ;

T =

Gc T; cT
;
Fc T; cT

where t will be specified below as part of the solution (note that
of a backward equation). We can think of Y as a Lagrangian with

; t

is now a solution

as the shadow price of

agent utility (reflecting both the participation constraint and incentive effect of consumption
on effort). Denote the optimal policy and corresponding agent utility (solving (5:18)) by
95

^
c; ^ U ; U , and write the dynamics of Y at the optimum as
^
dYt = Y dt + Y 0 dBt ;
t
t

YT = G T; cT + T F T; cT
^
^

^
Apply Ito’s lemma and substitute Y = V + t ^ U + Ut
t
t
t
Y = G t; c ; Y
^t t
t

^
t Ut

^U
t t

K ; Y
t

K

^
Ut

K

K :

(5.20)

t to get, for t 2 [0; T ),
t

^
Ut

K

t
(5.21)

0 ^ U + F t; c ; I t; c ; U ; ^ U ; U ; ^ U
^
^t
^t ^t t
t
t t
t
n
o0
Y
^
+
Ut K
^ ^ ^U
t
t I t; ct ; Ut ; t :
t

Theorem 5.3 below shows that optimality follows if for any other viable plan c; U and
corresponding U calculated from (5:18), we have for t 2 [0; T ):
Y
t

G t; ct ; Yt

t (Ut

K) ; Y
t

U
t t

(Ut

K) t

(Ut

K) t
(5.22)

0 U + F t; c ; I t; c ; U ; U ; U ; U
t
t
t t t
t t
t
n
o0
Y (U
U
+
t K) t I t; ct ; Ut ; t :
t
t

Note that the optimality conditions jointly specify the optimal principal policy c; ^ U ,
^
, as well as Y .

the corresponding Lagrangian drift process

The following Theorem provides the sufficient condition for principal’s problem under
less restrictive condition than Theorem 5.2 (see Remark 5.5 below).

Theorem 5.3. Assume the participation constraint binds. Let c; ^ U
^
cipal’s plan such that Y; Y ; ;

^
;U

be a viable prin-

solves the following FBSDE system for some
96

:
^
dUt =

F t; ct ; et ; Ut ; ^ U dt + ^ U 0 (dBt
^ ^ ^
t
t

et dt) ;
^

(5.23)

^
cT = F 1 T; UT ;
^
Gc T; cT
^
;
Fc T; cT
^

d t = t dt + t 0 dBt ;

T =

dYt = Y dt + Y 0 dBt ;
t
t
et = I t; ct ; Ut ; ^ U ;
^
^ ^
t

YT = G T; cT + T F T; cT
^
^

where Y satisfies (5:21). If (5:22) holds for any other viable c; U

K ;

then c; ^ U
^

is

optimal.

Proof. See the appendix.

Remark 5.5. Observe that the proof of Theorem 5.3 works if we assume concavity of
G (t; c; ), t 2 [0; T ), G (T; ) and F (T; ). These concavity assumptions are less restrictive
than the concavity assumption on G and F imposed to obtain sufficiency in Theorem 5.2.

The rest of this section restates the optimality condition of Theorem 5.3 and then reconciles it with the FOC in Theorem 5.2. Finally we illustrate with two examples.
Define

Y
t = Yt ; t ; t ; t

~
and, for any c; ~ ; U 2 Rd+2 ,
~

~
H t; c; ~ ; U ; t = G t; c; Yt
~
~

~
t U

K ; Y
t

t

~

~
+ t F t; c; I t; c; U ; ~ ; U ; ~
~
~ ~
n
o0
Y
~ K
+
U
~ ~ ~ :
t
t I t; c; U ;
97

~
U

K

t

Then Theorem 5.3 shows that optimality of c; ^ U is implied if, for all t 2 [0; T ],
^
~
H t; c; ~ ; U ; t
~

H ct ; ^ U ; U t ; t
^ t ^
all

t

~
U

^
Ut + t 0 ~

^ U , (5.24)
t

~
c; ~ ; U 2 Rd+2 :
~

Given any t and Ut , (5:24) can be used to solve for ct ; ^ U ; t as functions of ( t ; Ut )
^ t
^
^
(the resulting U then solves (5:18)). Given ct ; ^ U ; t and Ut we obtain
^ t
Y = H c ; ^U ; U ;
0 ^U
^
^t t ^t t
Ut K t
t
t t .
The resulting linked FBSDE system characterizes the solution to the principal’s problem.
~
If H is differentiable and concave in c; ~ ; U then optimality is implied if ct ; ^ U ; t
~
^ t
satisfy
Hc t; ct ; ^ U ; Ut ; t = 0;
^ t ^

H

t; ct ; ^ U ; Ut ; t = t ;
^ t ^

(5.25)

HU t; ct ; ^ U ; Ut ; t = t :
^ t ^
Remark 5.6. The FOCs for an interior solution which was achieved by using the utility
gradient approach are (recall that Fe =

U)
t

0 = Gc (t) + t Fc (t) + V 0 Ic (t) ;
t
V0
t = t F (t) G (t) + t I (t) ;
0
V0
t G (t) + I (t) + t IU (t) ;
t = t FU (t) GV (t)

(5.26)

~
which are identical to (5:25). Also note that concavity of H in c; ~ ; U is implied by the
~
concavity of F and G assumed in Condition 5.2 (b).
The following example shows how the result in [9], with expected utility for terminal
consumption only, is obtained in our setting. Example 5:5 which follows, shows a simple
98

extension to a recursive principal utility.
Example 5.4 (Terminal consumption only). Suppose there is no intermediate consumption
and the penalty for agent effort is quadratic:
1 0
qe e;
2

F (t; c; e; ) =

F T; cT = f cT ;

G (t; c; V; ) = 0;

t < T;

G T; cT = g XT

for some q > 0. From (5:5) optimal agent effort is et =

cT ;

U =q, and (5:26) implies the
t

following dynamics of :
e
d t = t 0 dBt ;

T =

g 0 X T cT
:
f 0 cT

Optimal agent volatility is

V =q
t
t which implies the key simplification dVt = qd t ,

and therefore Vt

for some constant . Substituting the terminal conditions for

V and

q t=

gives
= g XT

cT

g 0 X T cT
f 0 cT

q

!

;

which can be used to solve implicitly for optimal cT as a function of

(5.27)
and XT . To solve

for , apply Ito’s lemma to ut = exp (Ut =q) to get dut = ut e0 dBt and therefore (using
t
u0 = eK )
!
f cT
:
exp (K) = E exp
q
The martingale representation theorem gives the optimal effort e.

Example 5.5. Suppose the agent’s utility is the same as in Example 5:4, but the principal’s
is of the recursive form

G (!; t; c; ) =

1 V 0
q
;
2

G T; cT = g XT
99

cT ;

where q V > 0. Defining the ordinally equivalent transformation vt =
Ito’s lemma implies
v0 =
The

n

q V g XT

cT

o

q V Vt ,

:

of Example 5:4 therefore applies after replacing
n
o
exp
q V g ( ) , and the optimality condition (5:27) becomes

solution

with g ( ) =
~

=

5.5

E0 exp

exp

exp

n

q V g XT

cT

o

"

1 + aq V

g 0 X T cT
f 0 cT

!#

g( )

:

Translation-Invariant Preferences

Section 5:4 characterized the solution to the principal problem as an FBSDE system. The
solution is complicated by the links among the utility and Lagrange multiplier processes.
Under translation-invariant (TI) preferences, the Lagrange multiplier process is a constant
and the solution dramatically simplifies to a two-step problem: first solve a single unlinked
BSDE (which yields the optimal principal plan), and then plug the optimal plan into a single
forward SDE. The solution can be constructed in a straightforward manner, and optimality
easily confirmed.
[44] introduce TI recursive utility as a generalization of time-additive exponential utility. TI utility has the tractability of the latter, but allows more flexible modeling of risk
aversion and intertemporal substitution. For the optimal portfolio choice problem, [44]
show that the solution under TI preferences reduces to the solution of a single backward
equation, even in the presence of constrained markets and non-traded income. We show
below that the same simplification is achieved in the principal-agent problem.
We assume throughout this section that the set of intermediate consumption plans is
e
C = L2 (R). Next we define TI preferences:
100

Definition 5.5. We will say that the agent’s and principal’s preferences are TI if

F (!; t; c; e; U; ) = f

!; t;

G (!; t; c; V; ) = g !; t;

c
U

U; e;

X (!; t)
V

;

c

V;

;

G (!; T; c) =

for some constants U ; V > 0 and some functions f :
g:

[0; T ]

c
;
U

F (T; c) =

[0; T ]

X (!; T )
V

c

;

R1+2d ! R and

R1+d ! R. The process X 2 L (R) is interpreted as the principal’s cash

flow process.

The dependence of the principal’s aggregator on X is convenient for our applications,
but neither extends nor restricts the generality because the aggregators are still allowed to
depend on !. For any given effort process e 2 E, it is easy to confirm the quasilinear
relationships
Ut c + k U ; e = Ut (c; e)+k;

Vt c + k V ; e = Vt (c; e) k;

for all c 2 C, k 2 R;
(5.28)

where U and V solve the BSDE’s (5:1) and (5:2), respectively. The tractability of the TI
class all derives from (5:28).
Let
xU =
t

ct
U

Ut ;

X
ct
xV = t
t
V

Vt :

(5.29)

Redefining I from (5:6), the agent’s optimal effort is given by et = I t; xU ; U
t
t
d+1 ! Rd solves
I:
[0; T ] R
n
o
I (!; t; x; ) = arg max f (!; t; x; e; ) + 0 e ;
e2E
101

t 2 [0; T );

where

and the utility processes under agent optimality are therefore
o
n
dt + U 0 dBt ,
f t; xU ; I t; xU ; U ; U + U 0 I t; xU ; U
t
t
t
t
t
t
t
t

dUt =

(5.30)

c
UT = T ;
U
n
dVt =
g t; xV ; V
t
t

+ V 0 I t; xU ; U
t
t
t

o

dt + V 0 dBt ;
t

VT =

XT

cT
V

:

Because optimal effort is invariant to constant shifts in consumption (which follows because the agent’s utility diffusion is invariant to such shifts), the quasilinear property is
preserved.
The following examples give some special cases of agent TI preferences, each with an
aggregator separable in consumption, effort and volatility. The effort-penalty function is
given by

:

[0; T ]

Rd ! R (typically assumed convex in e), and the volatility

penalty is assumed either zero or quadratic. (The case of the principal is analogous, but
with no effort penalty.)
Example 5.6 (risk-neutral agent). Suppose the agent’s aggregator is

f (!; t; x; e; ) = x

(!; t; e) ;

2 R:

Then time-t agent utility is expected (under P e ) discounted future consumption minus
effort penalty:

e
Ut = Et

8
<Z T
: t

e

= U (s t)

cV
V s

(s; es ) ds + e

9
U (T t) cV =
=
T :
V;

The following example obtains, as a special case, time-additive exponential utility with
an effort penalty in the discounting term.

This is the class examined, for example, in

[27], [41] and [33]. The example also shows that recursive utility in the special case of
102

no intermediate consumption and quadratic volatility penalty is equivalent to time-additive
utility.

Example 5.7 (additive exponential with no intermediate consumption). Suppose

f (!; t; x; e; ) =

q 0
2

(!; t; e) ;

Then the ordinally equivalent utility processes ut =

q > 0:

exp ( qUt ) satisfies (under suffi-

cient integrability)

ut =

e
Et

(

exp

"

Z T

c
q T
U

t

(s; es ) ds

#!)

:

That is, u is standard additive exponential utility with coefficient of absolute risk aversion
q= U , and with effort affecting utility through the change of measure and the discounting
of terminal consumption by the effort penalty.

The next example is the special case of additive exponential utility with intermediate
consumption:

Example 5.8 (additive exponential). Suppose

f (!; t; x; e; ) =

exp ( x)

Then the ordinally equivalent process ut =

1 0
2

(!; t; e) :

exp ( Ut ) has the solution (under sufficient

integrability)

ut =

8
> RT
>
< t exp
e
Et
>
> + exp
:

cs
U
cT
U
103

Rs
t (r; er ) dr
RT
(r; er ) dr
t

9
>
ds >
=
>
>
;

:

5.5.1

Optimality

The following lemma shows the key simplifying property of the TI class: that the Lagrange
multiplier process

is a constant, equal to the ratio of the

parameters (the same result

can be deduced from the dynamic programming result in Theorem 5.3).5
Lemma 5.2. Assume Condition 5.2 (a). Under the TI aggregators of Definition 5:5, at the
optimum (c; e)
U
t= V;

t 2 [0; T ] :

In particular, the participation constraint always binds.
Proof. On the one hand, letting "0 = (1; )0 , Lemma 5:1 and Theorem 5:2 imply6
"V fVt (c + )
lim t
#0

Vt (c)g + "U fUt (c + )
t

Ut (c)g

= 0:

On the other hand, by quasilinearity (5:28), the left-hand side above equals "U = U
t
"V = V . So we get t = "U ="V = U = V . The participation constraint binds because
t
t t
= 0 = U = V > 0 (see Theorem 5:2).
The fact that the participation constraint binds is an intuitive result because constant
shifts in consumption do not affect optimal agent effort. Any consumption plan resulting
in a slack constraint could be improved upon by reducing the agent pay by a small constant
process. In view of this we will pursue the dynamic programming approach of Section
5.4.2.
As explained in Section 5.4.2, the principal solves the pay process by choosing the
intermediate consumption stream fct ; t < T g, as well as the agent utility diffusion process
U . This implies that lump-sum terminal consumption is c = U U , where U is the
T
T
T

5 Note that the proof of Theorem 5:4 doesn’t rely on the lemma, and therefore doesn’t
assume Condition 5.2.
6 It is trivial to extend Lemma 5:1 to obtain the time-t gradient needed here.
104

terminal value of the SDE

dUt =

o
n
dt + U 0 dBt ;
f t; xU ; I t; xU ; U ; U + U 0 I t; xU ; U
t
t
t
t
t
t
t
t

(5.31)

U0 = K:
To motivate the optimality result in Theorem 5.4 below, we suppose c; ^ U is the optimal
^
plan, and (similar to Section 5.4.2) define for this plan the linear combination
U
Yt = Vt + Ut ;

where

=

V

;

which satisfies
dYt = Y dt + Y 0 dBt ;
t
t
Using the identity xV =
t
Y = g t;
t

x U + Xt = V
t

xU + X t = V
^t

YT = XT = V :

(5.32)

(Vt + Ut ) we get

Yt ; Y
t

^ U + Y 0 I t; xU ; ^ U
^t
t
t
t

+ f xU ; I t; xU ; ^ U ; ^ U ;
^t
^t
t
t
c
^
^
Ut :
where xU = t
^t
U

(5.33)

t 2 [0; T );

A dynamic programming argument implies that for any other viable plan c; U the drift
term is smaller:
Y
t

g t;

xU + Xt = V
t

Yt ; Y
t

U + Y 0 I t; xU ; U
t
t
t
t

+ f xU ; I t; xU ; U ; U ;
t
t
t
t
c
where xU = t
Ut :
t
U

105

t 2 [0; T );

(5.34)

Theorem 5.4. Assume the TI aggregators of Definition 5:5: Suppose Y; Y

solves the

BSDE (5:32) where Y solves (5:33) for some viable c; ^ U . Then c; ^ U is optimal
^
^
if and only if (5:34) holds for any other viable plan c; U .
Proof. See the appendix.

It follows from the theorem that optimality is essentially equivalent to solving the Bellman equation
Y =
t

x U + Xt = V
t

max
fg t;
U ; U 2R Rd
xt t

U + Y 0 I t; xU ; U
t
t
t
t

Yt ; Y
t

+ f xU ; I t; xU ; U ; U g;
t
t
t
t

all t 2 [0; T ):

(5.35)

That is, optimality reduces to maximizing, for each (!; t), the negative of the drift of the
linear combination of the aggregators. Writing the maximizing arguments of (5:35) as
xU =
t
:

t; Xt = V
[0; T ]

Yt ; Y
t

U =
t

and

Rd+1 ! R and

:

[0; T ]

BSDE for Y; Y :

dYt =

n
g t;

x U + Xt = V
t

+ Y 0 dBt
t

YT = XT = V ;
xU =
t; Xt = V
t

Yt ; Y
t

t; Xt = V

Yt ; Y
t

Yt ; Y for some functions
t
d+1 ! Rd , we obtain the following
R

U + f t; xU ; e ; U
t
t t t

I t; xU ; U dt ;
t
t

;

U =
t

t; Xt = V

Yt ; Y
t

o

dt (5.36)

.

If the maximization problem in (5:35) is well defined, and the BSDE has a unique solution,
then optimality of xU ; U follows.
The BSDE does not depend on either the agent or principal’s utility process or utility diffusion. If f is a deterministic function (that is, f depends on ! only through its
106

other arguments), then so is I, and if g is also deterministic, then X is the only source of
uncertainty driving Y .
Let Yt (X) denote the solution to (5:36) corresponding to the cash-flow process X.
This solution can be interpreted as a subjective time-t present value of the cash-flow process;
a present value that depends on the preferences of both principal and agent. It is easily seen
that Y inherits the quasilinearity property of the principal and agent, but with respect to the
cash-flow process instead of consumption:
Yt X + k V

= Yt (X) + k;

for all k 2 R and t 2 [0; T ] :

It follows that the optimal xU ; e is invariant to constant shifts in X, and, from the SDE
(5:31), so are the utility process U and the optimal consumption plan c. Therefore any
constant unit shift in the cash flow process all accrues to the principal, whose utility process
increases by 1= V .
Solving the optimal principal utility under TI preferences in our moral hazard problem is therefore equivalent to solving a simpler TI utility problem with given consumption process X and modified TI aggregator. The solution immediately yields the optimal
xU ; U , which can then be substituted into the forward equation (5:31) to solve for the
agent’s utility process, U , and the optimal consumption plan
n
o
U xU + U ;
ct =
t
t

cT = U UT :

(5.37)

The following example shows that additively separable agent and principal aggregators
implies that Y also has a separable form.

Example 5.9 (separable absolute aggregators). Suppose the aggregators are separable in x
107

and (e; ):
f !; t; xU ; e;

= hU !; t; xU + k U (!; t; e; ) ;

g !; t; xV ;

= hV

!; t; xV

Optimal agent effort takes the form et = I t; U
t

(5.38)

+ k V (!; t; ) :

where I :

[0; T ]

Rd ! Rd .

Defining, for all t 2 [0; T ),
H t;

Xt
V

Yt

Q t; Y
t

= max
xU 2R
t
=

then the BSDE for Y is

dYt =

H t;

hV
8
> V
< k

max
U 2Rd >
:
t

Xt
V

t;

xU +
t
t; Y
t

Yt + Q t; Y
t

Xt
V

Yt + hU t; xU
t

;

U + k U t; I t; U ; U
t
t
t
+ Y 0 I t; U
t
t

dt + Y dBt ;
t

YT = XT = V :

(5.39)
9
>
=
>
;

;

(5.40)

Example 5.10. If we assume (5:38) with the added restriction that hU = hV (common
rankings of deterministic consumption plans), then, denoting by h the common aggregator
function, the optimum is xU = (1 + ) 1 Xt = V
t

H t;

Xt
V

Yt

Yt and therefore

= (1 + ) h t; (1 + ) 1

Xt
V

Yt

:

The next example shows that if either principal or agent exhibits infinite elasticity of
intertemporal substitution (h is affine) then H in (5:40) is affine (the case of no intermediate
consumption, which corresponds to hU and hV depending only on (!; t), is a special
case).
Example 5.11 (infinite elasticity). Let the aggregators take the separable form (5:38). We
108

show that if either principal or agent’s aggregator is affine in intermediate consumption
(i.e., either exhibits infinite elasticity of intertemporal substitution):
hi !; t; xi =

(!; t) + (!; t) xi ,

for i = U or i = V;

2 L R+ ;

(5.41)

then
H (!; t; x) =
for some process

(!; t) + (!; t) x

(5.42)

2 L (R), which we now specify. Optimality of the first equation in

(5:39) is equivalent to (if i = U then

i denotes V , and vice versa) t = hx i t; xt i

and inverting we get xt i = t for some 2 L (R). If i = U then t =
U
hV (t; t ). If instead i = V then t = t
t t + h (t; t ).

5.5.2

t

t t+

Quadratic Penalties

We specialize the TI preferences to the case of quadratic volatility and effort penalties:
f !; t; xU ; e;

g !; t; xV ;

= hU !; t; xU + pU (!; t)0

= hV

!; t; xV

where Qe ; QU ; QV 2 L Rd d

+ pV (!; t)0

1 0 U
Q (!; t)
2

1 0 e
e Q (!; t) e;
2
(5.43)

1 0 V
Q (!; t) ;
2

are assumed symmetric positive definite, and repre-

sent the effort and risk-aversion penalties; and pU ; pV 2 L Rd can be interpreted as
differences in beliefs of the agent and principal from the true probability measure. We
can interpret pU , for example, as a measure of agent optimism in the sense that under the
U
agent’s subjective probability measure P p the drift of the Brownian increment dBt is
U
pU
pU dt (that is, dBt = dBt pU dt is standard Brownian motion under P p ).
t
Recall from Example 5:8 that the case of additive exponential utility is the special case
with pU = pV = 0 and QU = QV = I, where I is the identity matrix, and hU and hV
109

are exponential functions. Let us define weight matrix W 2 L Rd d as
Wt =

Qe
t

Lemma 5.3. The BSDE for Y; Y

dYt =

H t;

1

1 + QV + QU
t
t

Xt
V

Yt

Qe
t

1 + QV
t

t 2 [0; T ] ;

;

given the aggregators (5:43) is

+ Y + pY 0 Y
t t
t

1 Y0 Y Y
Qt t
2 t

dt + Y 0 dBt ;
t
(5.44)

YT = X T = V ;
where H is defined in (5:39) and
QY =
t
pY = (I
t

1 QU W
t t

1 ;

Qe
t

0
Wt )0 pV + Wt pU ;
t
t

Y =
pU
t
2 t

pV
t

0

(I

Optimal agent effort satisfies et = Qe
t

Wt ) QU
t

1

pU
t

pV
t

:

1 U . The optimal xU satisfies the first equation
t

in (5:39), and the optimal agent utility diffusion is
U =
t

1W

Y
t t + (I

Wt ) QU
t

1

pU
t

pV
t

:

(5.45)

Lump-sum terminal consumption is given by cT = U UT , where UT is the terminal value
of the SDE

dUt =

hU t; xU + pU U
t t

1 U0
U
t Qt
2

Qe
t

1

U
t

dt + U 0 dBt ;
t
(5.46)

U0 = K:
110

Proof. See the appendix.

Y = V +
U , then
t
t
t
(5:46) in the case pU = pV gives the optimal volatility-sharing rule7 V = (I Wt ) Y
t
t
U = Y , t 2 [0; T ]. If the agent is risk neutral, QU = 0, then W = I and all the
and
t
t
t
t
Recalling that the subjective cash-flow volatility satisfies

time-t risk is optimally borne by the agent. If the principal is risk-neutral and agent effort
is infinitely expensive, QV = Qe
t
t

1 = 0, then W = 0 and the principal bears all the
t

time-t risk.
In the case of scalar penalties we obtain below a convenient expression for the lumpsum terminal pay, as well as simplified comparative statics.8

Example 5.12. Suppose QU = q U I, QV = q V I, Qe = q e I, where I is the d d identity
t
t
t
matrix and q e ; q U ; q V 2 R+ . Then Wt = wI and QY = q Y I where
t
w=

1 + qeqV
;
1 + qeqV + qeqU

qY =

1

1
qe

qU w

;

(5.47)

and optimal effort satisfies

et =

1 w U
w Y
e t + q e q U pt
q

pV
t

:

(5.48)

The coefficient of Y in (5:48) is decreasing in q e and q U , but increasing in q V . We can
rearrange (5:44) to obtain an expression for

Y0
t dBt , and substitute into (5:46) together

7 Note that both W and I W are positive definite for all (!; t).
t
t
8 See the appendix for a derivation of the example.
111

with U = q e et , with et given by (5:48), to get9
t

cT = w X T

+ U

K

E XT + V

Z T
0

Z T
0

X
H t; t
V

Yt

X
E H t; t
V

qU

dt

(5.49)

!

hU t; xU dt

Z T
2
w Y
1 w
+
+
dt
p U pV
t
t
t
2
qU
0
Z T
Z T
Y 0 pU dt + U 1 w
V
w
pU p V
dBt
t t
t
t
U
q
0
0
8
2
3 9
>Z
>
qY Y 0 Y
Y + pY 0 Y
> T
>
<
6
7 =
t
t t
2 t t
6
7 dt :
+ Vw
> 0 4
qY Y 0 Y 5 >
Y + pY 0 Y
>
>
:
;
E
t
t t
2 t t
U

Yt

!

1
qe

pU dt
t

The first component of the agent’s terminal pay is a fixed share of the unexpected component of terminal cash flow, as well as a share of the unexpected cumulative transformed
intermediate consumption. The second compensation term is the participation constraint
adjusted for utility of intermediate agent consumption. When principal and agent agree
(pU = pV ), the third term increases terminal agent pay proportional to the quadratic variation of Y , which in typical models is driven by uncertainty about the cash flow process.
The next two terms represent adjustments for agent optimism as well as an additional disagreement term. The final term compensates for the unexpected cumulative drift in the
process Y . The proportion w (from (5:47)) is decreasing in q e and q U but increasing in
q V . The more risk averse the agent and the more costly the effort, the greater the share
is needed to incentivize effort. A more risk averse principal will optimally keep a smaller
share of the risky cash flow.

9 Recall U = c = U , Y = V + K, and xV = X = V
t
0
0
T
T
t
112

Yt

xU .
t

Before presenting the main example of this section, we give the solution to the BSDE
(5:44) in the special case when H is affine and QY = 0.

Example 5.13 (QY = 0). Suppose the conditions of Example 5.11 (infinite elasticity),
which implies the affine form of H in (5:42), and suppose QY = 0 for t 2 [0; T ]. Then Y
t
is obtained via risk neutral discounting:
8
>ZT R
<
s
pY
Yt = Et
e t u du
>
:
t

Xs
Y
s V + s+ s

9
>
RT
=
du XT :
t u
ds + e
V >
;

(5.50)

When H is affine (see Example 5:11), the BSDE (5:44) (after substituting (5:42)) is

essentially the same as the BSDE (21) in [44] (which applies to the optimal portfolio problem). They provide sufficient conditions on the BSDE parameters and Markovian statevariable processes such that the Y will be an affine function of the state variables, the coefficients of which satisfy a set of Riccati ODEs. The following simple example considers
a one-dimensional state variable representing the cash-flow process X.

Example 5.14. Assume that the cash-flow process satisfies the forward SDE

dXt =

X

XX

t dt +

d
X

i=1

Xi

q
i
ai + bi Xt dBt ;

0
for some X ; X ; bi 2 R+ , and ai , Xi 2 R, i = 1; : : : ; d. Let X = X1 ; : : : ; Xd .
Suppose hU and hV are deterministic functions, and assume the conditions in Example
5.11 which imply the affine form for H in (5:42); for simplicity we assume

is constant.

Also, suppose all the preference parameters (QU ; QV ; Qe ; pU ; pV ) are deterministic. The
solution for Y in the two cases below is affine in the state variable:

Yt = 0 (t) + 1 (t) Xt ;
113

(5.51)

where the deterministic processes 0 and 1 are given in closed form below.

a) (Ornstein-Uhlenbeck dynamics) Let ai = 1 and bi = 0, i = 1; : : : ; d. Then
X+

1
1 (t) = V e

0 (t) =

Z T
t

e

(T t)
0

(s t) B
@

+

V

X+

(

1

X+

e

(T t)

)

;

1
Y0 X
X+
Y +
s+ s
C
1 (s) ps
1 (s)
A ds:
1
(s)2 X0 QY X
s
2 1

Substituting Y = 1 (t) X , then optimal agent utility diffusion is deterministic:
t
U =
t

1W
X + (I
t 1 (t)

As cash-flow mean reversion blows up,
we get

1

Wt ) QU
t

X ! 1, then

pU
t

pV
t

:

U
V
1 ! 0 and, if p = p ,

U ! 0; that is, the impact of effort on cash-flow drift becomes more transient,

and the optimal contract transfers no cash-flow risk to the agent.. If there is neither mean
reversion nor intermediate consumption,

=

X = 0, then

V and therefore
1 = 1=

Y = X= V .

b) (Square-root dynamics) We further assume unbiased beliefs (pV = pU = 0), and
constant diagonal preferences (Qi = q i I, q i > 0, i 2 fU; V; eg). Using the notation (5:47)
t
we have
r+ 1= V
1 (t) =

0 (t) =

1= V
Z T
t

e

r

e

r

e
0

(s t) @

1 (s)

r+ r
r+ r

(T t)

1= V

(T t) 1= V

1 Y
X+
2
s+ q
1 (s)
2
114

r+ r
;
r+
d
X

i=1

1

Xi 2 ai A ds;

where (assuming the expression in the radical is positive)
X+
r

=

r

X+

2

+ 2 q= V

q = qY

;

q

d
X

Xi 2 bi :

i=1

Substituting
Xi
Yi =
1 (t)
t

q
ai + b i X t ;

then optimal agent utility diffusion is
Ui =
t

1w

Xi
1 (t)

q
ai + b i X t ;

i = 1; : : : ; d:

and therefore effort, et = (q e ) 1 U , is increasing in the cash-flow process. Terminal
t
consumption is

cT = w X T

E XT +

Z T
0

Z
d
q Y V X Xi 2 T
w
2
0
i=1
Z
U
1
w 2 T
U
+
q
2
qe
0

fXt

!

E (Xt )g dt

2 i
1 (t) b fXt
d
Xn

Xi

i=1

+ U

K

Z T
0

!
U t; xU dt
h

E (Xt )g dt

o2
(t)
ai + bi Xt dt:
1

Terminal consumption is a fixed fraction w of terminal cash flow plus a cumulative weighted
average of the cash flow over the agent’s lifetime. The last term on the right represents
compensation for risk bearing, which could be negative if q e is small enough (inducing a
desire for risk by the agent), and includes a stochastic component compensating the agent
for the volatility of the cash-flow process over the life of the contract when b is nonzero.

115

For the special case of no intermediate consumption ( =
8
>
>
<

5.6
5.6.1

1 (t) = >
>
:

e

= 0) we have
1

X (T t) V
X (T t)
q
+ X e
2
n
o
V + q (T t) 1
2

if

X 6= 0

if

1

X =0

Appendix
Proofs

We will start by presenting a Comparison Lemma for BSDE. It is based on [4]. The following lemma is a variation of their result with a similar proof.
Lemma 5.4 (Comparison). Suppose U i ; i , i 2 fa; bg, solve the BSDE
i
dUt =

i
f i t; Ut ; i dt + i0 dBt ;
t
t

where f i :

[0; T )

process (t)

0 and two constants

R

i
UT = f i (T ) ;

Rd ! R and f i (T ) :
0;

! R. Assume that there exist a

> 0 such that f i ; i 2 fa; bg satisfies the

following conditions,
a) for all (t; ; u; u0 ) 2 [0; T ]
f i (t; u; )

b) for all (t; u; ) 2 [0; T ]

Rd+2 ;
f i (t; u0 ; )

u

u0 ;

Rd+1 ;

f i (t; u; )

(t) + juj +

2

k k2 ;

RT
c) 0 (s)ds , f i (T ) has exponential moments of all order;
d) f i (!; t; u; ) is concave for all !; t; u 2
[0; T ) R:
116

i 2 fa; bg ;

If
f a (t; u; )
a
then Ut

f b (t; u; ) ;

(t; u; ) 2 [0; T )

Rd+1 ;

f a (T )

f b (T ) ;

b
a
Ut , t 2 [0; T ]. If the inequalities (5:52) are reversed then Ut

Lemma 5.5. Suppose U; U

~
and U ; ~ U

(5.52)

b
Ut , t 2 [0; T ].

solve the BSDE (5:1) for some feasible

plans (c; e) and (~; e), respectively. If
c ~
F t; ct ; et ; Ut ; U + U 0 et
t
t
F T; cT
then Ut

F t; ct ; et ; Ut ; U + U 0 et ;
~ ~
t ~
t

t 2 [0; T );

(5.53)

F T; cT ;
~

~
Ut , t 2 [0; T ]. If the inequalities in (5:53) are reversed then Ut

Proof. Define the nonnegative process

~
Ut .

as the difference between the left and right-hand

sides of (5:53). From (5:1) we have

dUt =

n

o
+ F t; ct ; et ; Ut ; U + U 0 et dt + U 0 dBt ;
~ ~
~
t
t
t
t

UT = F T; cT :

n

o
+ F t; ct ; et ; Ut ; U + ~ U 0 et dt + ~ U 0 dBt ;
~ ~ ~
~
t
t
t
t

UT = F T; cT ;

Compare to
~
dUt =

and apply Lemma 5:4 (the required conditions of the lemma are satisfied by Definition 5.1;
note that F (t; c; e; U; ) is concave in (U; )).

Proof of Theorem 5.1
1) (Sufficiency) Lemma 5.5 implies that U0 (c; e)
therefore e is optimal.
117

U0 (c; e) for any e 2 E, and
~
~

2) (Necessity) Suppose e is optimal and (5:4) is violated for some e 2 E. Let
8
>
U
U0
< e if F t; c ; e ; U ; U + U 0 e
t
t t t t
t t F t; ct ; et ; Ut ; t + t et ;
et =
~
>
: et
otherwise.

By Remark 5.1 it follows that e 2 E: Then (5:53) holds with the inequalities reversed (the
~
inequalities are strict). Lemma 5.5 implies U0 (c; e) < U0 (c; e). But this contradicts the
~
supposed optimality of e.

Proof of Theorem 5.3
Consider any viable plan c; U , and let V = V (c) 2 E and Jt = Yt t (Ut K) 2
^
E . We will show that V0 J0 , with equality holding if c; U = c; ^ U . The terminal value is

JT = G T; cT + T F T; cT
^
^

F T; cT

= G T; cT + T

where (using the concavity of G and F in c, and T > 0)
^
T = G T; cT
Gc T; cT
^

G T; cT + T F T; cT
^
cT
^

F T; cT

cT + T Fc T; cT
^

cT
^

cT

= 0:

Now define t , t 2 [0; T ), as the nonnegative process which is the difference between left
and right-hand sides of (5:22). Applying Ito’s lemma we get

dJt = dYt
Substituting J = Y
t
t

t (Ut

t dUt

(Ut

K) d t

d t dUt :

K) + t F t; ct ; et ; Ut ; U
t
118

t

0 U , using definition
t

of t and (5:22), we get
n

dJt =

J
t + G t; ct ; Jt ; t

JT = G T; cT + T :

o

dt + J0 dBt
t

I t; ct ; Ut ; U dt ;
t

Comparing to the BSDE

dVt =

G t; ct ; Vt ; V
t

dt + V 0 dBt
t

I t; ct ; Ut ; U dt ;
t

VT = G T; cT

V0 c; U , with strict equality holding for

and applying Lemma 5:4 implies that J0

the optimal plan. (Note that a binding participation constraint implies J0 = Y0 .)

Proof of Theorem 5.4
Sufficiency: This is a special case of Theorem 5.3; however we give a self contained
proof for completeness. Consider any viable plan c; U , and let V = V c; U denote
the principal’s utility (from the solution V; V

to (5:30)). Define Jt = Yt
Ut . We
^
J0 , with equality holding if c; U = c; ^ U . Define nonnegative

will show that V0
process

as the difference between left and right-hand sides of (5:34). Substituting the

dynamics of Y and U into dJt = dYt
dJt =
JT =

t + g t;
XT

cT
V

Xt

ct
V

dUt , together with J = Y
+ J0 I t; xU ; U
t
t
t

Jt ; J
t

U , yields
dt + J0 dBt ;
t

:

Comparing to the BSDE

dVt =
VT =

g t;
XT

cT
V

Xt

ct
V

Vt ; V
t

+ V 0 I t; xU ; U
t
t
t

;

119

dt + V 0 dBt ;
t

V0 cV ; e , where et = I t; xU ; U , with
t
t

and applying Lemma 5.4 implies that J0
strict equality holding for the optimal plan.
c; ^ U
^

Necessity: Let
c; U . Define

be the optimal viable plan.

Consider some viable plan

as the difference between left and right-hand sides of (5:34). Suppose
[0; T ] that belongs to F

that t < 0 on some subset of

B[0;T ] with a strictly positive

l (where l is the Lebesgue measure on [0; T ]) measure. Define, for t 2 [0; T ),

P

xU ; ~ U
~t
t

=

8
>
<

xU ; U
t
t
xU ; ^ U
^t
t

>
:

if t < 0;
otherwise;

and et = I t; xU ; ~ U . By Remark 5.1 it follows that (~; e) 2 C
~
~t
c ~
t
SDE

n

~
dUt =

o
U ; e ; ~ U + ~ U 0 e dt + ~ U 0 dB ;
f t; xt ~t t
~
t
t ~t
t

~
~
~t
and define ct = U xU + Ut . Then Jt = Yt
~
~
dJt =

min (0; t ) + g t;

Xt

ct
~
V

~
E: Let U solve the
U0 = K;

~
Ut satisfies the BSDE
~
Jt ; ~ J
t

+ ~ J0 et dt + ~ J0 dBt ;
t ~
t

X
cT
~
~
;
JT = T
V
~
where cT = U UT . Comparing to the BSDE
~
~
dVt =

g t;

Xt

ct
~
V

~
Vt ; ~ V
t

+ ~ V 0 et dt + ~ V 0 dBt ;
t ~
t

~
~ c ~
Lemma 5.4 implies J0 < V0 (~; e). That is, Y0
et = I t; xU ; ^ U , which contradicts optimality.
^
^t
t

X
cT
~
~
VT = T
;
V

~ c ~
K = V0 (^; e) < V0 (~; e), where
c ^

Proof of Lemma 5.3
Substitute f

=

QU U + pU , g

=

QV V + pV , and Ie = Qe into the second

120

FOC in (5:26) (recall U = I =

fe ) to get (omitting time subscripts throughout)

1 V =

Qe

QU U

QV V

p U + pV

and therefore
V =

Substitute (5:54) into Y = V +
Y =

Qe

1 + QV

1

QU U

pU + pV

:

(5.54)

U to get
pU

pV +
1 U
Q

Substituting I + [Qe ] 1 + QV

U =

1

1 + QV

Qe

1 W Y + (I

1 + QV

Qe

I+

1 U
Q

U:

= W 1 and solving for U we get

W ) QU

where we have used the identity W [Qe ] 1 + QV

1

1

pU

= (I

pV

(5.55)

W ) QU

1

. Substi-

tuting the same identity into (5:54) we get
V = (I

W) Y +

W 1 (I

W ) QU

1

QU (I

1

W ) QU

pU

I

pV

and therefore
V = (I

W) Y

(I

W ) QU

121

1

pU

pV

:

(5.56)

Substituting e = (Qe ) 1 U , the dynamics of Y are (omitting the arguments to H)
8
>
<

dY =

1
H + pV 0 V + pU 0 U 2 V 0 QV V
n
o
U 0 (Qe ) 1 + QU
U + Y 0 (Qe ) 1 U
2

>
:

Substituting (5:55) and (5:56) we obtain
pV 0 V + p U 0 U = pY 0 Y +

pU

pV

(I

W ) QU

U

2 Y 0 Qe

1

9
>
=
>
;

dt + Y dB:

pU

pV

(5.57)

and10
1n V0 V V
Q
+
2
1 Y0 Y Y
=
Q
2
2

pU

pV

0

U0

QU

n
1

Qe

1 + QU

W )0

(I

where
QY = (I
+

8
>
<

QV

W)

1 QU W

1 Qe

Qe

9
>
=

> + (Qe ) 1 + QU >
:
;

W )0 QV (I W )
n
o
1 W 0 Qe 1 + QU W

= QV (I
=

o

1 Qe

1W

(I

1 U

o

(5.58)

W ) QU

1

1 W 0 Qe

1

pU

1W

1 ;

where the last equality is easily confirmed (and the symmetry of QY is also easily con10 It is easy to confirm that the cross term of the form Y fg pU pV (and its transpose) is zero using
n
o
(I W )0 QV W 0 Qe 1 + QU + Qe 1 = 0:
122

pV

;

firmed).
Finally, substitute11
QU

1

W )0

(I

n

o
1 + QV + QU (I

Qe

W ) QU

n
o
e ) 1 + QV + QU (I
(obtained using the identity (Q

1

= QU
1

W ) QU

1

(I

W )0

= I) into the

last term of (5:58) and add to (5:57) to get
Y =

5.6.2

2

pU

pV

W ) QU

(I

1

pU

pV

:

Derivation of Examples

Derivation of Example 5.12
Substitute optimal agent utility diffusion
U =
t

1

1w Y +
t

w
qU

pU
t

pV
t

into the agent’s SDE (5:46) and integrate to get

UT

8
>
Z T>
<

K=

+

Z T
0

0 >
>
:
1

hU t; xU

w
qU

pU
t

1 U
q
2 t
pV
t

dBt

1w Y
t
dt
w
>
>
+ 1U
pU pV
;
t
t
q
Z T
1w
Y dB
U
U dt +
pt
t pt dt :
t
0

e
qt

11 Note that the following implies symmetry of (I
123

9
2>
>
=

1

W ) QU

1

.

Now integrate the BSDE for Y to obtain
Z T
0

Y
t

dBt

pU dt =
t

8
Z T>
<

Xt
H t; V

0 > + (1
:

+ XT = V

w) pV
t

Yt + Y
t
0 Y
1 qY Y 0 Y
pU
t
t
t t
2

Y0 ;

9
>
=
>
;

dt

and substitute, along with cT = U UT , and (from (5:44))
Z T(
X
H t; t
Y0 = E
V
0

qY
2

Y
Yt + Y + pt 0 Y
t
t

Y0 Y
t t

)

dt + E XT = V

to get the result.

Derivation of Example 5.14
Define i = Xi
1 (t) Xt to get

q
ai + bi Xt , i = 1; : : : ; d, and apply Ito’s lemma to Yt =

0 = _ 0 (t) + _ 1 (t) Xt + 1 (t)
+ Y + 1 (t) pY 0
t
t

X

XX

t + t+ t

Xt
V

0 (t)

0 (t) +

1 (t) Xt

1
(t)2 0 QY :
t
2 1

For the case QY = q Y I and pY = 0 (the extension to nonzero pY and nondiagonal QY
t
t
when b = 0 is obvious), the resulting Riccati system is

0 = _ 0 (t) + 1 (t) X + t + Y
t

0 (t)

d
X
1 Y
2
q
1 (t)
2
i=1

Xi 2 ai ;

0 (T ) = 0;
0 = _ 1 (t) +
V
1 (T ) = 1=

1 (t)

X+

V:
124

d
X
1 Y
q
(t)2
1
2
i=1

Xi 2 bi ;

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125

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