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CONTINUOUS TIME ARBITRAGE APPROACHED AS A PROBLEM IN
CONSTRAINED HEDGING

By

James Andre Demopolos

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1999

ABSTRACT

CONTINUOUS TIME ARBITRAGE APPROACHED AS A PROBLEM IN
CONSTRAINED HEDGING

By

James Andre Demopolos

I characterize absence of arbitrage with tame portfolios in a model where a finite
vector of stock prices is symbolized by a continuous semi-martingale with respect to the
completed filtration generated by a vector-valued standard Brownian Motion. Levental
and Skorohod (1995) solved this problem using probabilistic methods in the sub—case of
invertible volatility matrix. They constructed an arbitrage trading strategy based upon
domination at the end of the time horizon of the value of one stochastic process by that of
another. This construction through domination suggests a link between the arbitrage
problem and the mathematical theory of financial hedging of contingent claims. This
dissertation does not assume invertible volatility. In the case of singular volatility, one
faces the constraint that the dominating process constructed by Levental and Skorohod
cannot always be effectively converted into a process symbolizing the accumulated
capital gains of a trading strategy. Therefore, to apply the theory of hedging, one must
consider hedging under constraints. This dissertation contains two primary results. First,
I generalize Levental and Skorohod's characterization of arbitrage opportunities in terms
of a domination relationship between stochastic processes. Second, I apply work by
Cvitanic and Karatzas (1993) pertaining to hedging with constrained portfolios to this
generalization to provide a new characterization of absence of arbitrage in the case of

singular volatility. The proofs are probabilistic. Some examples are provided.

To my wife, Emily, and my daughter, Sofia

iii

- ACKNOWLEDGEMENTS

I would like to express my gratitude to the co-chairmen of my dissertation committee.
Professor Levental's knowledge and pleasant demeanor were of great value in the
completion of this research, and I am grateful for the opportunity to study probability
from as accomplished a mathematician as Professor Skorokhod. I also owe sincere
thanks to several other members of the faculty in the Department of Statistics and
Probability. I thank Chairman Salehi for helping me in many ways during my stay at
Michigan State, Professor Melfi for providing consistently excellent instruction in
theoretical probability, Professor Stapleton for giving courses characterized by his
diligence and insight, and Professor Page for her guidance during my experience as a
consultant. Professors Erickson, Gilliland, LePage and Fabian also contributed to my
successful completion of this program, and I duly thank them. I would also like to thank
Cathy Sparks for her assistance with administrative matters, and Alexander White for

useful discussions early in the research.

iv

TABLE OF CONTENTS

Introduction

Setting and Main Results
1.1 The Model. . . . .....................
1.2 The Link Between Arbitrage and Constrained Hedging. .......

1.3 The History of the Arbitrage Problem. ..............

Immediate Arbitrage
2.1 Preliminaries. ........................

2.2 Proof of the Immediate Arbitrage Theorem. ............

A Construction from the Theory of Constrained Hedging
3.1 The Construction and Closely Related Properties. ..........
3.2 An Essential Lemma in the Link to the Arbitrage Problem. ......

3.3 Two Useful Characterizations of the Process V(-) ...........

Characterizations of Arbitrage and Corollaries
4.1 Characterization of Arbitrage in Terms of Domination of Z(T; 1) .....

4.2 Characterization of Arbitrage in Terms of the Processes V(t; o).
Examples

Bibliography

10

16

22

22

24

30

3O

36

4O

43

43

53

66

76

INTRODUCTION

A basic problem in the construction of asset price models in mathematical finance is
the determination of the conditions which are necessary and sufficient for a specified
model to exhibit the absence of arbitrage, i. e., the absence of risk-free profit
opportunities. In this work, a characterization of absence of arbitrage is provided in the
context of a specified model for a finite vector of stock prices. The work has been
motivated by results in Levental and Skorohod (1995) and Cvitanic and Karatzas (1993).
The former paper characterizes absence of arbitrage in a restricted version of the model
considered here. As will be explained in detail in Chapter 1, Levental and Skorohod's
Corollary 3 [page 920] suggests a link between the problem of characterizing absence of
arbitrage and the theory of hedging contingent claims. In the more general setting of this
dissertation, fewer stochastic processes can be taken to meaningfully symbolize
accumulated discounted capital gains than in Levental and Skorohod's work. This
limitation motivates consideration of the problem of hedging under constraints in the
course of attempting to provide a hedging based approach to the arbitrage problem. The
problem of hedging contingent claims with constrained portfolios is precisely the topic of
Cvitanic and Karatzas (1993). The approach of this dissertation is to generalize
Corollary 3 of Levental and Skorohod to provide a characterization of arbitrage
"reminiscent of hedging" in the setting of this work, and then to apply the framework of
Cvitanic and Karatzas to this generalization in order to provide a new characterization of
absence of arbitrage.

The paper is organized in the following way:

In Chapter 1, I specify the model, and motivate the link between the theory of hedging
with constrained portfolios and the arbitrage problem. The dissertation's major results are
stated. Chapter 1 also contains a history of research into the arbitrage problem.

Chapter 2 contains the statement and proof of necessary and sufficient conditions for
the absence of a special kind of arbitrage, namely, immediate arbitrage. Loosely
speaking, in an immediate arbitrage, an investor does not ever let his capital gains
become negative in the process of obtaining almost sure positive capital gains at the end
of the time horizon. Although I did have to make modifications to the argument, the core
of the proof, particularly on the necessity side, appears in Levental and Skorohod (1995)
[Lemma 2, page 914.]

In Chapter 3, I adapt the work of Cvitanic and Karatzas to this arbitrage problem. The
details differ from their work in that in this paper, I need to impose constraints which
depend upon (t,oo), whereas their constraints do not vary with (Loo).

Chapter 4 accomplishes the stated objectives of this work. Theorem 3 is the promised
generalization of Levental and Skorohod's Corollary 3. Theorem 4 results from
application of the constructions derived from Cvitanic and Karatzas' work in Chapter 3 to
the characterization given in Theorem 3. Theorem 5 extends the conclusions of
Theorem 4 to address the issue of the equivalence of absence of arbitrage and the
existence of an absolutely continuous local martingale measure for the stock price
processes.

Chapter 5 contains examples. As will be explained herein, Examples 1 and 4 show
that the characterization of arbitrage in the setting of this work is meaningfully different

from Levental and Skorohod's (1995) characterization.

Chapter 1

Setting and Main Results

1.1 The Model.

Consider a financial market in which one bond, with price process B, and d 2 1 stocks,
with price processes SI, ..., Sd, are traded in the time interval 0 S t S 1. Unless otherwise
specified, all processes herein will be taken to be defined for O s t S 1. Correspondingly,

in definitions of stopping times interpret the infirnum of an empty time set as 1. The

source of uncertainty in the market is a d-dimensional standard Brownian Motion
W = (W1, W2, ..., Wd)* defined on a complete probability space (Q, F, P)l. The term

“adapted” will refer throughout to the filtration {F :2 0 s t S l}, the P augmentation of the

natural filtration of W, namely
(1.1) ‘ Ftéo{W(s):‘OSsSt}vU

where U = {A e F: P(A) = O}. The price processes of the financial instruments evolve

according to the equations

(1.2) I dB(t)¥B(t)r(t)dt, B(O)=1.

 

' * will denote matrix transpose.

(1.3) dS.(t) = Smi 26.,.(t)de(t) + brow],

lede

Si(O)=Si E(0,00), 1:1,...,d.

Here r(-) is an adapted R-valued process symbolizing the instantaneous force of interest,
volatility 0(a) is an adapted d x d matrix-valued process not necessarily invertible for any
(I, (D), and drift b(-) is an adapted Rd-valued process.

In order that (1.2) and (1.3) have well-defined solutions, we require that

(1.4) KW)! +Z|b,(t)| +Zo§j(z)}dt < 00 as.

i

A continuous-time trader chooses a portfolio, namely, the amount of money to invest in

each of the d stocks at each time t. Formally define a portfolio by

Definition 1. A portfolio is an adapted Rd-valued process it which satisfies the

integrability constraint
1 1‘) u

(1.5) prsmsyl‘+in'(s)a(s)[}ds < oo a.s.,
0

where // // denotes the Euclidean norm in Rd_ and with
Id =(1, 1, I)“ 6 Rd, the process a is defined by
(1.6) a(t) = b(t) — r(t)1d.

Since (1.4) implies that the paths B(-) satisfy inf{B(t) : O S t _<_ 1} > O a.s., constraint (1.5)

implies that the semi-martingale X,t in Definition 2 below is a well-defined process.

Definition 2. The process X7t given by
(1.7) X, (z) = {3“ (s)(1t '(s)c (s))dW(s) + {13“ (5)1: ‘(s)a(s)ds
o o
is the discounted capital gain process associated with the portfolio 1t.

To motivate Definition 2, begin from the purpose of investing in stocks, namely, the

attempt to obtain capital gains in excess of those available from the less risky bond. In

this light, n‘(t)o(t)dW(t) + 1t.(t)b(t)dt — r(t)«n‘(t)1d dt =

 

 

_ no) _ 11,0)
- :[Sj(t)dsj(z) 80) (13(1)),

which is verbally,
{instantaneous gains from portfolio investment in nj(t) / SJ-(t) stock shares, j = 1, ..., d} —
{opportunity cost of foregone instantaneous gains possible from the bond}.

Multiplying by B’l(t) discounts these excess (or deficient) gains from stock investment to
their present value at time 0. Integration across time sums the discounted instantaneous
gains.

Common wisdom is that a reasonable model for the processes Sj and B should not

allow for risk-free profits. This-is the no arbitrage principle.
Definition 3. An arbitrage is a portfolio 1: such that the associated discounted
capital gain process X7t satisfies
1) There exists a C > 0 such that P{Xn(t) 2 —Cfor all 0 S t S 1} = 1.

ii) P{X.(1) 2 0} = 1.

iii) P{X.(1) > 0} > 0.

Any portfolio 7: for which the associated Xn satisfies i) in Definition 3 is called a tame
portfolio. C-tameness means that i) is satisfied with respect to a particular C. Tameness
is a restriction that prevents “doubling schemes” and can be interpreted as putting a limit
on borrowing. The mathematics underlying "doubling" in continuous time was set forth
by Dudley (1977), who showed that in our model with d = 1, an arbitrary F 1 measurable
random variable A (including, in particular, A satisfying A > O as.) can be represented as

1 l
A = Ig(t)dW (t) for an adapted process g satisfying Ig2(t)dt < 00 as. In his
0 0

construction, it is possible that for each C > O,

f
P{min0$t$1 Ig(S)dW(S) < “C} > O.
0

The relationship between the absence of as. positive capital gains and the requirement of
tarneness is treated rigorously in Dybvig and Huang (1988) [see Theorem 2, page 390.]
The purpose of this dissertation is the study of conditions equivalent to the absence of
arbitrage. To that end, we need define numerous objects. Let it denote Lebesgue
measure on [0, 1]. As Shreve has shown, unless there is a projection-based arbitrage,

then we must have

(1.8) X®P{a(t, 0)) e Ran[o(t, co)]} = 1.

[See Karatzas and Shreve (1998), Theorem 1.4.2, page 12.] Condition (1.8) will be

assumed throughout this workz. It holds for each (t,(o) that

Ran[o(t,w)] = Ran[o(t,(u)o'.(t.(o)] and that 00.0)) is injective on Ran[o.(t,co)]. Therefore,
we may uniquely (up to a HEP null set) and adaptedly define a relative risk process 9
such that 9(t,(u) e Ran[o‘(t,a))] for all (1.0)) and o(-)9(o) = a(-) X®P a.s.3 Using 9,

define a stopping time or:

t+h

(1.9) a = inft>02 ]||9(s)||2ds :00 for all he(0, 1-1].
1

a is a legitimate stopping time because of right-continuity of the Brownian filtration, and

is the key object in a characterization of the absence of a special kind of arbitrage.
Definition 4. An immediate arbitrage is a portfolio 11; for which there exists a
stopping time 0 S t S 1 satisfiing P{‘C < 1} > 0 such that

P{Xn(t) = Ofor all t S t anan(t) > 0for all t > t}: 1.

Theorem 1 (Immediate Arbitrage Theorem.) There is no immediate arbitrage if

andonly ifP{0t =1} = 1.

The primary contribution of this work pertains to characterization of arbitrage when

immediate arbitrage does not exist. Therefore, as is consistent with Theorem 1, for the

 

2 See Chapter 2, Proposition 1, for the details of the necessity of ( l .8) for the absence of arbitrage.

3 To define 6 such that it is an adapted process, define 9(t) = 0:] (t) a(t), O S t S l, where 6+ is an
adapted process such that for each (t,a)), o+(t,(o) is an invertible dxd matrix and for each x e Ran[o(t,a) )],
o+"(t,m) x e Ran[o'(t,u))] and o(t,m) o.'l(t,w) x = x. The existence of such an adapted o. is proven in
Lemma 1 of Chapter 2.

remainder of this chapter all results will be given under the assumption that a = l as. In
the absence of immediate arbitrage, the fundamental objects in results about existence of
arbitrage are exponential local martingales. For each adapted Rd-valued process v
satisfying orv = 1 a.s., where stopping time av is defined as in (1.9) with process v
substituted for process 9, define for each stopping time 0 S t S 1 another stopping time

C(r) by

t
(1.10) C(r) = inf+t>02 [{z<s}{iv(s)|]3ds = 00}.
0

(Adopt the convention of denoting the indicator function of a set by the set itself.) Using

CV“), define an adapt?d process 2v“; ') by

t 1 t . v
(1.11) Zv(t;t) = exp{ git <S}V (S)dW(S) - 36“ T < Q ill"(5)“ st} 1f t < Z; (T)
0, if t> CV (15).

Z‘ (I; Q" (1)): liminf tVTC ( )Zv(‘t;t).

We have that lim Zv (t ;t) exists as. The limit exists on

T; “m
{lfiht<slv(s)||2ds <OO}C{QV(T)=1}
O

1
because “I: < s}v '(s)dW(s) is well-defined on this set and the stochastic integral with

0

respect to Brownian Motion has continuous paths. The limit exists and equals 0 as. on

{hr < sfi!v(5)”2d5 = 00} 2 {6(7) < 1 },

because if

t 1
W0[ [{r < s}fiV(s)H2dsJ = fir < s}v*(s)dW(s),
0 0

then

1
{WOOL t < I{t < s < §V(t)}||v(s)!|2 d5}
0

is a standard Brownian Motion in R1 [see Karatzas and Shreve (1981), page 174], and so

1
(1.12) 11m {Too ‘W0(t) - 11/2 = —00 3.5. on {fit < SH|V(S)"2dS = (13} -
0

Observe that (1.12) implies that the paths Z"(r; .) are continuous as.
The term exponential local martingale is appropriate because if CT?) = 1 a.s., then

that ZV(T; .) is a local martingale follows from that for each t, on {t < §V(t)}
(1.13) Z"('r;t) = l — [{t <s}Z"(t;s)v'(s)dW(s).
0

It is also relevant that Z"(r; .) is a nonnegative supermartingale for any v and t for which

the process is defined. Observe that for each t, Z"(t; t) = ZVCIZ; t A §V(t)) as. The

integral representation (1.13) implies that there exist stopping times CK“) S C” (t) ,

n 2 1, such that limnT Q: (1:) = Q" (t) and Zv(r; . A§;(t)) is a martingale for each n. So

if 0 S s < t, the F atou Lemma for conditional expectation implies that a.s.,
(1-14) 5127901175] = E[ lim. Z“(t;t ACX(T))1F.1 -<- limn E[T(t;tA€Z (1))11’. ]

= limn Z"(t;s./\ gm) = Zv(r ;.s/\§"(t)) = Z"(t ;s).

Since Z"(t; .) is a superrnartingale, it holds that ZV(T; .) is a martingale if and only if
E(Zv(t; 1)) = 1.

Since the process 9 is of central importance, simplify notation by denoting for each I
the process 29(1; .) merely as Z(t; o). Abbreviate Z"(O; .) as Zv(o) for any adapted

process v satisfying or" = 1 as. So Z(-) will denote 29(0; o).

1.2 The Link Between Arbitrage and Constrained Hedging.

The importance of the processes Z(t; -) in the arbitrage problem has been studied by
many. Levental and Skorohod (1995) proved that in the absence of immediate arbitrage,

under the additional assumption that s(t,w) is invertible for all (t,co), that absence of
arbitrage is equivalent to E(Z(r; 1)) = 1 for all constant times 0 S r S 1. Their proof uses

in a substantial way the invertibility of o. This work extends that of Levental and
Skorohod in that it removes the assumption of invertible volatility, allowing

7t<8>P{o(t,o)) is singular} > 0. Of fundamental importance in Levental and Skorohod's
proof that an arbitrage exists if there exists a stopping time t such that E(Z(t; 1)) < 1 is

the existence in that case of an exponential local martingale Z‘”(-) which satisfies

10

P{Z‘p(l) > Z(t; 1)} = 1. In fact, their Corollary 3 [page 920] states that in the case of

‘dt < 00 a.s., that the existence of

 

 

I
invertible o, with the added assumption that HP (t)
0

arbitrage is equivalent to the existence of an adapted Rd-valued process (p satisfying

1
fll@(’)ll2df < 00 as. such that
0

(1.15) P{Z“’(1)2Z(1)} =1 and P{Z“’(1)>Z(l)} >0.

Equation (1.15) suggests a link between the arbitrage problem and the theory of
hedging. To understand this link, begin with consideration of a "seller's objective in
hedging." [See Karatzas (1996), Section 0.4 for more detail than is given here] Define a
contingent claim to be a non-negative Fl-measurable random variable". One can view a
contingent claim as a financial obligation at time 1 to which a seller commits himself in

exchange for money at time 0. Let A be a contingent claim. For each x 2 0 such that

there exists an adapted Rd-valued process it such that

 

1 2
(1.16) [{Haana)” + n*(r)b(z)]}dt < 00 a.s.,
O
l 1
(1.17) x + ]n*(r)o(t)dW(z) + ]n*(t)b(z)dr 2 A a.s.,
0 0

and there exists a constant C > 0 such that we have the tameness constraint

 

‘ Typically in work focusing on hedging, additional constraints which imply absence of arbitrage are
assumed to apply with respect to process 6. A requirement related to these additional constraints is then
included in the definition of a contingent claim [see Karatzas (1996), page 10]. Since mention of hedging
is intended to be motivational here, and since I have not assumed absence of arbitrage, I have chosen to
omit these additional details.

11

I I
(1.18) P minoggl ]n*(s)o(s)dW(s) + 1r*(s)b(s)ds < —C = 0,
0 0

we have the interpretation that a seller can "hedge" his obligation to pay A at time 1
starting with the purchase price x at time 0.5 Examination of (1 .13), the integral
representation of process Z‘p(-), suggests the link between (1 . 15) and this "seller's
objective in hedging." Consider an "auxiliary market" in which asset price processes are
characterized by the original invertible volatility 0', but the drift b is replaced by the zero

vector process. Then define an adapted Rd-valued process rt by

no) = [o‘(t)1"(-Z“’(t><p(t)).

it satisfies (1.16) for the "auxiliary market," since the paths Z“’(-) are continuous and we

1
have fll(p(t)||2 dt < 00 as. for the process (p in (1.15). Because
0

t . .
]n*(s)o(s)dW(s) = Z‘p(t) — 1; 0 St s 1,
o .

(1.15) implies that (1.17) holds with x =1 and A = Z(1). (1.18) holds with C =1.
Loosely speaking, we may conclude that the existence of arbitrage is the same as the
existence of a portfolio which hedges Z(l) starting from initial wealth 1 in an "auxiliary

zero drift market."

 

5 The left hand side of (1 . 17) symbolizes initial wealth plus non-discounted capital gains through time 1.
Since the seller is obligated to pay A at time 1, not at time 0, it would be inappropriate to discount the
capital gains to their present value at time 0 here.

12

The following theorem is a generalization of Levental and Skorohod's Corollary 3. It
gives a similar characterization of the existence of arbitrage in terms of domination at
time 1 without integrability constraints applied to 9 beyond or = 1 as. o is not assumed

to be invertible.

Theorem 3. Assume absence of immediate arbitrage. Arbitrage exists if and only If

there exist both an adapted Rd -valued process (p satisfying
P{or‘p = 1} = 1 and 7t®P{cp(t,(u) e Ran[o‘(t,(o)]} = 1

and a stopping time 0 S t S 1 such that processes Z‘Wr; .) and Z(t; .) are not

indistinguishable and P{Z‘p(r; I) 2 Z(‘C,’ 1)} = 1.

Here, the additional condition of primary importance beyond those stated in the
characterization of arbitrage in Levental and Skorohod's Corollary 3 is the range
requirement, (p(t,o)) e Ran[o*'(t,m)] X®P a.s. Recall that in the preceding discussion
linking Corollary 3 to the concept of hedging, the portfolio which hedges Z(l) in the
"auxiliary zero drift market" is a linear function of the process (p for which
P{Z‘”(l) 2 2(1)} = 1. Therefore, this additional range condition suggests that research
into the problem of "hedging with constrained portfolios" may be useful for this text's
arbitrage problem. In their 1993 paper, Cvitanic and Karatzas give a control theoretic
characterization of the minimal initial wealth level required for a seller to hedge a
contingent claim under the constraint that the hedging portfolio must take values for all
(t,m) in a fixed convex subset of Rd. The range condition in Theorem 3 places a similar

constraint on process (p, although the convex set varies with (t,03). In fact, dependence on

13

(t,0o) of the set Ran[o'(t,00)] is not an insurmountable obstacle to application of the work
of Cvitanic and Karatzas. This dissertation's primary accomplishment lies in bridging
their results and Levental and Skorohod's approach to the arbitrage problem.

To incorporate Cvitanic and Karatzas' theory of hedging with constrained portfolios
into a solution of the arbitrage problem, we need to introduce various objects. Let D

denote the class of all adapted Rd-valued processes v(-) satisfying (1.19), (1.20) and

(1.21):

(1.19) 70®P{v(t,c0) e Ker[c(t,a))]} = l.
1

(1.20) P{ J“v(s)“2ds < co} =1.
0

(1.21) E(ZV(1)) = 1.

For each v e D, let Pv denote the probability measure on ((2, F) with Radon-Nikodym
derivative dPV/dP = 2(1). Let Ev (EV[ o | F , ] ) denote expectation (conditional
expectation “given F , ” ) with respect to P”.

For each stopping time 1:, define for each t e [0, 1] random variable Vo(r; t) to be a

version6

(1.22) V0 (1; t) = ess sup E V [Z (1:;1)|E 1
veD

 

6 By “ess sup” it is meant the following: If {X : is l } is a collection of random variable measurable
with respect to a o-field G, where the index set 1 is of arbitrary cardinality, then there exists an as. unique
G measurable extended random variable Y taking values in (-oo, oo ] which satisfies the following two
conditions:

(i) For each i e I, Y _>_ X, as.
(ii) If Y' is a G measurable extended random variable satisfying
(1), then ‘1' 2 Y a.s

Denote Y = ess sup{X, : is I }.

14

For each I, the adapted process Vo(r; .) admits a cadlag modification V(r; o), which by
right-continuity is unique up to indistinguishability. Use the abbreviation V(o) to denote
the process V(O; .), The processes V(r; .) and Z(t; .) are the central objects in this work's

characterization of arbitrage:

Theorem 4. Assume absence of immediate arbitrage. Then there is no arbitrage if
and only if processes Z(r; .) and V(r; .) are indistinguishable for all constant times

0SrSl

Theorem 5 below is an equivalent formulation of Theorem 4. In a sense, Theorem 4 is
a characterization of absence of arbitrage in terms of a stochastic supremum, while
Theorem 5 recasts this result in terms of an attained maximum. Theorem 5 is an
important tool in addressing the problem of equivalence of absence of arbitrage and the
existence of a probability measure Q << P such that the asset prices S;(-), i = 1, ..., d are
local martingales with respect to (O, F, {Ft}o 5 1s 1, Q). Worthy of note is that the proof of
Theorem 5 contrasts with approaches in the literature to the problem of existence of an
absolutely continuous local martingale measure in that it does not rely upon functional

analysis.

Theorem 5. Assume absence of immediate arbitrage. There is no arbitrage if and

only if for each constant time 0 S r S 1 there exists a 11 e D such that E (26 “Yr; 1)) = 1.

15

In the final chapter, I give examples illustrating the properties of the processes V(r; .)
and their relationship to the processes Z(r; o). Example 1 demonstrates that the
equivalent condition for absence of arbitrage in Theorem 4 is not equivalent to
E(Z(r;1)) = 1 for all constant times r. If there is no immediate arbitrage, then the latter
condition implies absence of arbitrage, but Example 1 serves as a counter-example to the
reverse implication by exhibiting both E(Z(1)) < 1 and no arbitrage. Example 2 shows
that we cannot simplify Theorem 4 by reducing the conditions equivalent to absence of
arbitrage to the behavior of the processes V(r; .) at time 0: in Example 2, we have
V(O) = 1 a.s., but V(-) and Z(-) are not indistinguishable. Although the processes V(r; .)
have cadlag paths, it is not true in general that they have continuous paths. In Example 3,
P{V(1/2) at 11mm aV(t)} > 0. Example 4 is due to Delbaen and Schachermayer (1998b).
It is similar to Example 1, which I constructed before discovering their paper. In

Example 4, there exists a v e D such that V(-) and E"[Z(l) I F.] are indistinguishable.

1.3 The History of the Arbitrage Problem.

Since the late 1970's researchers have actively investigated the question of which
properties of possible asset price models correspond to the absence of arbitrage
opportunities. Most of the resulting articles have focused in one way or another on the
notion of an equivalent martingale measure, namely, a probability measure Q equivalent
to the measure P such that the discounted asset price processes (denoted Si(o)/B(o),

i = 1, ..., d in this text's notation) are martingales on the original filtered probability space

with measure Q replacing P. That the existence of such a martingale measure is a

16

sufficient condition for absence of arbitrage in a wide variety of circumstances was
established early in research on this topic. For discrete time asset price models defined
on a finite probability space with finitely many time values, Harrison and Kreps (1979)
showed that the existence of such an equivalent martingale measure is necessary and
sufficient for the absence of arbitrage. An early result for continuous time trading models
appeared in Harrison and Pliska (1981); therein, the authors show that with a discounted
price model that is a cadlag strictly positive semi-martingale, the existence of an
equivalent martingale measure implies absence of arbitrage.

The question of whether absence of arbitrage implies the existence of a martingale
measure is complex, particularly in the case of continuous time process models. Almost
all proofs of the existence of a martingale measure have employed the Hahn-Banach
Theorem or one of its corollaries. In the discrete-time case, Harrison and Kreps (1979),
and similarly, Harrison and Pliska (1981) employed the separating hyper-plane theorem
to generate a linear fimctional symbolizing a pricing system with which one can construct
an equivalent martingale measure. Both of these papers worked with finite probability
spaces for the discrete-time problem. Taqqu and Willinger (1987) also established the
equivalence of absence of arbitrage and existence of an equivalent martingale measure
for a discrete time, finite probability space framework; their proof differed from
preceding works in that it used a geometric reformulation of the no arbitrage assumption.
Dalang, Morton and Willinger (1990) established the equivalence for discrete time
trading in general (non—finite) probability spaces. The equivalence for general probability

spaces was subsequently proved using‘somewhat simpler arguments than those in

17

Dalang, et. al. by Kabanov and Kramkov (1994) and Rogers (1995). Note that the
theorems referred to above all pertained to a finite number of trading times.

In discrete-time asset models, the issue for the infinite time horizon case is more
complicated. Back and Pliska (1991) provide an example allowing trading in the infinite
horizon which does not permit arbitrage, but for which there is no equivalent martingale
measure. The notion of "no fi'ee lunch," sometimes called "no approximate arbitrage," is
a stronger assertion than no arbitrage and becomes relevant here. There are several
formulations of "flee lunch" in the literature. Early definitions of the existence of "free
lunch" require a sequence of random variables, namely, terminal wealth levels for
discounted capital gain processes, to converge topologically to a nonnegative random
variable that is not as. 0. The topologies used to define the convergence vary by paper.
[See, for example, Kreps (1981).] Because a sequence of trading strategies which
require a trader to risk increasingly large losses, none of which produce probability one
positive capital gains, seems undesirable as an approximation to arbitrage, tameness
requirements were added to the definition of "free lunch." Schachermayer (1994) defines
the property of "free lunch with bounded risk" as the existence of a sequence of arbitrage
approxirnants which are each C-tame for a single C > 0. He proves that in the infinite
time horizon discrete trading problem, "no free lunch with bounded risk" (N F LBR) is
equivalent to the existence of an equivalent martingale measure. Furthermore, in the
infinite horizon discrete case, the need to prevent "doubling scheme" based arbitrage
becomes apparent. Harrison and Kreps (1979) provide an example of probability one
positive capital gains where the minimum value of the wealth process across time is not

bounded below as.

18

In continuous time trading, "doubling" based as. positive capital gains are possible in
a model admitting an equivalent martingale measure even with a finite time horizon.
Harrison and Pliska (1981) correct for this phenomenon by requiring tameness. Dybvig
and Huang (1988) provide a rigorous analysis of the impossibility of as. positive
discounted capital gains in a market admitting an equivalent martingale measure if one
adds the requirement of portfolio tameness. Regarding the problem of the existence of a
martingale measure, in the context of a continuous bounded semi-martingale model for
the discounted asset prices on time set [0, l], Delbaen (1992) proved the equivalence of
NFLBR and the existence of an equivalent martingale measure. The results of F ritelli
and Lakner (1995) include that under only the assumption that the discounted asset price
processes are adapted and right continuous, existence of an equivalent martingale
measure is necessary and sufficient for absence of "free lunch with stopping times." In
their work, the stochastic processes are defined on an arbitrary index subset of [0, co), and
"free lunch with stopping times" is defined as a sequence of arbitrage approximants for

which the portfolios processes lie in the linear span of
{1t(t) = g{t < t S [3}; t S B are stopping times, g e L°°(P), g is FF measurable}.

Duffie and Huang (1986) and Stricker (1990) study the relationship between "no free

lunch" and the existence of an equivalent martingale measure under the assumption that
the discounted asset price processes are in LP, 1 S p < oo. Duffie and Huang (1986) also

prove some interesting results about the relationship between "no free lunch" and the
relative sizes of filtrations generated by different agents' information. Delbaen and

Schachermayer (1994b) establish that if the discounted asset price process

19

{S(t); 0 S t < 00} is a bounded real valued semi-martingale, then there is an equivalent
martingale measure if and only if S satisfies "no free lunch with vanishing risk"

(NF LVR). NFLVR is defined to hold if for any sequence of positive constants 6n
satisfying limn 8n = 0, each sequence of Sn—tame portfolios an (where portfolios are

defined as predictable processes it for which the stochastic integral

S
{In(t)dS(t); 0 S s < 00} is well-defined and converges as. to a limit as 5—) 00) must
0

co

satisfy P-limn Inn(t)dS(t) = 0. As a corollary, they obtain that if S is a locally bounded
0

semi-martingale, then NFLVR is equivalent to the existence of an equivalent probability
measure under which S is a local martingale. This corollary complements Delbaen and
Schachermayer (1994a) in which examples are provided showing that for unbounded
continuous discounted price processes, NFLBR is not equivalent to the equivalent
martingale measure property.

The proofs of Delbaen and Schachermayer rely heavily upon functional analysis. For
the model in this thesis with invertible volatility, Levental and Skorohod (1995) prove
that an equivalent martingale measure exists if and only if there is "no approximate
arbitrage," a condition which means roughly the same thing as NFLVR. Their proof is
more probabilistic than that of Delbaen and Schachermayer. Levental and Skorohod
(1995) and Delbaen and Schachermayer (1995) both investigate the relationship between
the existence of an absolutely continuous measure Q << P under which the discounted
asset prices are martingales, and absence of arbitrage (as opposed to absence of "free

lunch") Levental and Skorohod (1995) use the martingale representation theorem to

20

show that in their model, assuming absence of immediate arbitrage, no arbitrage is
equivalent to the existence of an absolutely continuous probability measure Qr << P for

each 0 S r S 1 under which (with the expression given for the one-dimensional case)

{Sm/Ba), r .<. t < €90). {a} 1 , Q}

is a local martingale. Delbaen and Schachermayer (1995) show, referring back to the
(1994b) result proven using the Hahn-Banach theorem, that if {S(t); O S t < 00} is a
locally bounded semi-martingale , then absence of arbitrage implies the existence of an
absolutely continuous probability measure Q << P under which the discounted asset price
process is a local martingale. Delbaen and Schachermayer (19983) consider the case of
unbounded asset price processes. Assuming that {S(t); O S t < 00} is a semi-martingale,
they prove that NFLVR is equivalent to the existence of a measure Q equivalent to P

under which the discounted price process is a martingale transform, i. e.,
S
Icp(t)dM (t); 0 S s < 00 , where M is an Rd valued martingale, and (p is a predictable
0

M-integrable R+-valued process.

21

Chapter 2

Immediate Arbitrage

2.1 Preliminaries.

The following result, due to Shreve, demonstrates why we assume condition (1.8),
X®P{a(t, (1)) e Ran[0'(t, 03)]} = 1.

Proposition 1. If X®P{a(t, 00) e Ran[o(t, (1))]} < 1, then an immediate arbitrage
exists.

Proof. For each (t,(o), Rd = Ker[o"(t,0))] O Ran[o'(t,c0)], where EB denotes orthogonal
sum. Define an adapted R‘Lvalued process a1 by defining a1(t,00) to be the projection of

a(t,a)) on Ker[o‘(t,(o)]. Then define another adapted Rd-valued process it by

(2.1) Mt): 2 21.0 )
1+lla < )1

‘it is a portfolio. That it meets the integrability constraint in the definition of a portfolio
follows from that for each (t,co)
(2.2) 0‘1: 2 0 6 Rd and

211.3 = ”a,”2 so that 0 S n‘a <1.

Now define a stopping time ‘I.’ by

22

(2.3) 1: = inf{t > 0: 71({sza1(s)¢ 0} n [t, t + a] )> 0 for all e > 0}.

That I is a stopping time follows from right-continuity of {F(}. That (1 .8) does not hold
implies that P{t < 1} > 0. Furthermore, (2.2) yields that X, satisfies
P{Xn(t) = 0 for all t S r and Xn(t) > 0 for all t > r}=1.

So it is an immediate arbitrage. I

Let us now attend to some technical details used in the proof of the immediate

arbitrage theorem.

Lemma 1. Let 0' be an adapted dxd matrix valued process. Then there exists a (non-
unique) adapted process 6+ such that for each (I, (1)), 6+(t,0)) is an invertible dxd matrix
and for each x e Ran[o(t,to)] we have both 0.5! (t, (o)x e Ran[o'(t, (1)) ] and
oft, (9)0510, (1))x = x.

Proof. Let k(t,(u) = dim(Ran[o'(t,o))]) = dim(Ran[o(t,co)]), and let
{ej, fj, g], ; j = 1, ..., d}be a set of adapted Rd-valued processes such that for each (t,c0),
{e1(t,0)), ..., ek(t,w)}, {f1(t,oo), ..., fd-k(t,(0)}, and {g1(t,c0), ..., gd_k(t,0))} are bases for
Ran[o'(t,0))], (Ran[o*’(t,w)])i, and (Ran[o(t,0))])i, respectively. Take o.(t,c0) to be the
matrix representation of the full-rank linear map on Rd defined by

o+(t,(o)ej(t,a)) = o(t,c0)ej(t,0)), j = 1, ..., k(t,co),

0+(t,c0)f}(t,a)) = gj(t,o)), j = l, ..., d — k(t,oo).

That Ran[o(t,c0)] = Ran[o(t,co)o'(t,u))] justifies that 0+(t,0)) is invertible. The remaining

assertions made about 6+ are evident from its construction. I

23

In this chapter and the next, the Girsanov Theorem will be a useful tool. [See Karatzas

and Shreve (1991), Theorem 3.5.1, page 191.] For each adapted Rd-valued process v

I
satisfying fl|v(t)l]2 dt < 00 as, define another adapted Rd-valued-process
0

W’ =<w:',... w;')‘ by

t
(2.4) W,"(:) = Wi(t) + ]v,-(s)ds; 1SiSd.
0

If E(ZV(1)) = 1, and probability measure Pv is defined by dPV/dP = Z"(1), then

Wv is a d-dimensional standard Brownian motion on ((2, F, {Ft}, P").
1.2 Proof of the Immediate Arbitrage Theorem.

Theorem 1 (Immediate Arbitrage Theorem.) There is no immediate arbitrage if
andonly ifP{a = I} = 1.

Proof. (Necessity) Suppose that P(a < 1) > 0. Start the construction of an
immediate arbitrage by selecting a sequence of constants rk l 0 such that if stopping

times ak, k = 0,1,..., are defined by (1k: (0: + n.) A 1, then

(2.5) ;PHLfia,<1sa,_,}{]p(z)||2A-g;]dz gr} (1 {a<1}] < 00.

Get such constants satisfying (2.5) as follows: Let r0 = 1. After selecting

. {r}, i = 0, k—I} and defining {01), i = 0, k—I} as stipulated above, the divergence
1 7 1
limrw fin +r <t Sak_,)[|p (t)”" A —]dt = 00 as. on {or <1}
r
0

24

allows choice of 0 < rk < ‘/2 rk_1 such that
l

(2.6) P{{fia + r, <t ((11-1)[lp (0”? Alldt S1} {1 {or <l}] S 73;.
o rk

With the sequences {ark} and {rk} define a sequence of stopping times {Tk; k 2 l} by

rk

(2.7) t, = inf{t >0: {{a, <sSot,,.1)[|p(s)“2 A 1:!ds =1}Aa,_,.

The Borel-Cantelli Lemma implies that as. on {or < 1}, ‘tk < or“ for all k sufficiently
large.

Now fix c e (1, 2) and define two adapted scalar processes

/
oo 00

0(1): Z{ak<tsrk}k’c and r(t)= X109. <ISIklll9(’)|lA’[1/2)
k=1 k=1

Then it holds as. on {or < 1}that

so

 

[morons 2k“ 1).
(2.8) 13131 0 V, = lim——i-=—"——V—3 = iimQ—t—l—ll—nQ-W :00.
l t n-vao no n—rao c—
{humans} [2k]

Now define an adapted Rd-valued process 111 by

Bahia) .
——9 , 9 0
(2.9) 90) = “90)"2 0) ’f (”I

0, if 6(t)=0.

O S y S “0” implies that as.

 

(2.10) ”Mara? S [B2(t)y2(t)dt S ik’z‘ < 00.

So the stochastic integral in (2.11) and (2.12) below is well defined. If we put
(2.11) W{ JllV (5)“2dsJ = {v ‘(s)dW(s),
0 0

then W0 is a standard Brownian Motion in R1, so that
P{t'm’ won) —> 0 as t i 0} =1.
Then, observing that w (t) = 0 on {t < a} and that

2({111 (5):: 0}O(a, a+a])> 0 for alls> 0'

holds as. on {01 < 1}, we can conclude that

I

fw'(s)dW(s)
(2.12) lim 0 =0 as. on {oc<l}.

(la 1 “3
[ my (5)1123]

Using (2.10) and (2.12) for the inequality, and then (2.8), proves that as. on {on < 1}

 

 

 

iv‘ts)dW<s)+ Nomads ]B(s)12(s)ds
(2.13) lim’h 0 t O 1/3 _>_ lim’k I 0 1/3 = 00.
[ iliv (91%] [1020» was]

Let ‘I.’ be the stopping time

I

l , 1/3
(2.14) 1: = inf{t>or: Iii/’(s)dW(s) + Iul'(s)9(s)ds = [my (s)“2ds] }.

0 0

26

Expression (2.13) implies that as. on {0: < 1}, both ‘I.’ > a and ifa < t S r, then

(2.15) :fiu‘(s)dW(s) + :fw'(s)9(s)ds 2 “Malfdsj > 0.

Now observe that for an adapted scalar process g, 111 = g0, so that for all (t,(l)),
w (t,c0) e Ran[o.(t,(o)]. Therefore Lemma 1 implies that there exists an adapted process
(c‘). taking values in the invertible dxd matrices such that c‘(.)[(c‘).(.)]“w (.) = w (.).
Define an adapted vector process 7:0 by mm = [(ol)+(t)]"\y (t). Then define the
immediate arbitrage 7t by

tt(t) = B(t){t S r}7to(t).

Regarding integrability constraints on a portfolio, the paths B(-) are bounded in t a.s., and

(2.16) (IIB"(r)o'(r)n(r)||2+|B“'0:m'0)a<t>l}dt = ji’S'ClQlilmllz+lil'(f)9(t)|)1i

s J(B2(t)+B(t)){z(t)dt _<_ i(k’“+k“) a.s.

Since B“(.)n‘(.)e(.) = {. s t}ut‘(o), and x. satisfies the SDE

(2.17) dx.(t) = B“1 (t)0'(t)dW(t) + B"(t)e(t)e(t)dt, x.(0) = 0,
it follows from (2.15) that as.

(2.18) X,(t) = 0, ift s a and x.(t) > 0, ift> a.

So 71: is an immediate arbitrage with a serving as the stopping time required in the

definition.

27

(Sufficiency) Suppose or = 1 as. Let 7t(-) be a portfolio such that there exists a
stopping time ‘t for which
P{Xn(t)= O for all 0 StSt and Xn(t)> 0 for allt> 1} =1.

Define another stopping time B by

B = inf{t>0: ]{t<s}lp(s)||2ds 2 I}.

0

Note that at = 1 as. implies that ‘L' < B as. on {r < 1}. We also have Z(t; B) > 0 as.

Further observe that the Novikov Condition implies E(Z(t; B)) = 1 [See Karatzas and
Shreve (1991 ), Corollary 3.5.13, page 199.] Define an adapted vector process 9 by

B(t) = B(t){t < t s [3}.
Then the process Z9.(-) is Z(r; . A B), so that E( Z6 (1) ) = 1. Because
P{Xn(t) = 0 for t S r} = 1 and X7: satisfies the SDE (2.17), the Girsanov Theorem implies
that on ((2, F1, {F t}, Pei ), the process X.(- A B) is a stochastic integral with respect to
W6, a standard Brownian Motion in Rd [see (2.4)]. In particular, X.(- A B) is a (P9!)

local martingale. Then, since P6 is equivalent to P, we have

(2.19) P“ {X.(t) = 0 for all 0 s t St and x,(t) > 0 for allt> t} = 1.

Therefore, x..(.A 13) is a (P6 ) supennartingale.‘ Then 55(X,(8)) s afar, (1)) = 0, so

that P6 (x.(8) = 0) = 1 by (2.19). By probability equivalence, P(x.(8) = 0) = 1.

 

1 It follows from the Fatou Lemma for conditional expectation that any local martingale which is
bounded below is also a superrnartingale.

28

Sor < B as. on {I <1} andP{X,,(t)> O for allt> t} = 1 imply thatP{'t = 1} =1. Since

it and I were chosen arbitrarily, no immediate arbitrage exists.

29

Chapter 3

A Construction from the Theory of Constrained Hedging

3.1 The Construction and Closely Related Properties.

Throughout Chapters 3 and 4, assume absence of immediate arbitrage, i. e., assume
or = 1 as. The results in this chapter specialize work of Cvitanic and Karatzas (1992) and
(1993) to the problem of this dissertation. This chapter contains little that Cvitanic and
Karatzas did not prove. The usefulness of Corollary 1 is‘specific to this arbitrage
problem; so it did not appear in their work. All proofs, barring that of the existence of the
cadlag modification of V0 in Theorem 2 draw at least their key probabalistic content from
Cvitanic and Karatzas. They give a different proof, which also appears in El Karoui and
Quenez (1995), justifying the existence of the cadlag modification, but to the best of my
knowledge the one given here has not appeared elsewhere. Their papers constructed the
V process for the problem of pricing contingent claims in incomplete markets, and did
not recognize its usefuleness for the arbitrage problem. Comparison with their papers
will show that my approach is to'view Z(‘t’; l) as a contingent claim to be hedged in a
market with d risky assets characterized by our original volatility process 0' and drift

process identically zero.

30

The results and proofs of this chapter will use the notation V(-) and Z(-) otherwise
reserved for ‘t = 0. All of the proofs go through without alteration for V(r;-) given any
stopping time 0 S r S 1. I have chosen the simpler notation since nothing herein depends
upon I.

Recall that we define D to be the class of adapted Rd-valued processes v for which

(1.19) A®P {v(t,00) e Ker[ o(t,co) ]} =1,
1

(1.20) P{ J]|v(s)||2ds < co} = 1,
0

(1.21) E(ZV(1)) = 1.

For each v e D, Ev denotes expectation and conditional expectation with respect to the
probability measure Pv with Radon-Nikodym derivative dPV/dP = Z"(1).

Let V0 be an adapted Rl-valued process such that for each 0 S t S 1, Vo(t) is a version
of ess supveD (EV[Z(1)| Ft]). In working with essential suprema, the following lemma
will be of use.

Lemma 2. F or each stopping time 0 S ‘I.’ S I, there exists a sequence {vm' n 2 1} in D

for which we have the a. s. monotone convergence
(3.1) ess sup..o(E"[Z(1)| F.]) =11:an EV"[Z(1)| F.].

Proof. If a family of random variables is directed upward, then the essential

supremtun of the family is the as. increasing limit of a sequence in the family [see

31

Neveu (1975), Proposition VI-l-l , page 121.]1 So fix I and show that

{EV[Z(1) I F.]: v e D} is directed upward. Let v] and v; be in D and define

A = {EV1[Z(1)|F,] 2 .Ev2 [Z(1)|F,]} e F, .
Then define an adapted process 11 by
“(0 = Vl(t)[A fl {t> Tl] + V2(t)lAC I] {t > T}]-
u e D: it obviously satisfies (1.19) and (1.20). For (1.21), because A 6 F.,

F.])

Then, for j = 1, 2, using that (1 .20) implies P{min 0 5., , z"i(t) > 0} = 1, and optionally

(3.2) E(Z“(1)) = E(AE[ZV'(t;1)I~;D + E(A‘E[Z"2(t;1)

 

 

stopping martingale 2‘",

FT] = E 21(1)”; = 1 E[z"f(l)|F,] = 1.
Z"1'(t) zvf(t)

 

 

(3.3) E[z"f (1,1)

 

So E(Z“(1)) = P(A) + P(Ac) = 1. Since for each v e D,

(3.4) EV[Z(1) I F.] =E[Z(1) Z"<r;1) l 12.],
A e F. implies

(3.5) E“[Z(1) | F.] = AEV1[Z(1) | F.] + AC Ev2[Z(1) | F.]

2 Ev1[Z(1) | F.] v EV2[Z(1) | F.] as.

 

' A family H of random variables is said to be directed upward if for each h], h; e H, there exists an
I13 E H With 113 2 h] V 1']: 3.5.

32

Theorem 2. (i) For each v e D, V0 is a (PV) supermartingale.
(ii) V0 admits a (P) modification with cadlag paths.

(iii) Denote the cadlag modification, unique up to indistinguishability, guaranteed by

(ii) as V. If i; is another process with cadlag paths such that I} is a (F) supermartingale
for each v e D and 17(1) = Z(1)a.s., then

P{V(t) s tin) for all 0 s t s 1} = 1.

Proof. The (P‘) supermartingale pr0perty for V0 for each v e D will follow

immediately from the following identity, which holds for all 0 S r S s S 1.

(3.6) Vo(r) = ess supveD(E"[Vo(s) I F r]).

In (3.6), "S" stems merely from monotonicity of conditional expectation:

(3.7) Vo(r) = ess supveD EV[Z(1) I F,] = ess supveDEVIE‘IZU) I F.] I F,]
S ess supvep'EVst) I F,].

For "2", fix u e D, and prove Vo(r) 2 E“[Vo(s) | F,]. We know by Lemma 2 that there

exists a sequence {vn} ; D such that Vo(s) = lim'ln Ev"[Z(1) I F,]. For each n define an

adapted process un by

11.0) = 11108 s s} + vn(t){t > s}.

Each [in e D since, with a calculation like (3.3) justifying the third equality in (3.8),
(3.8) E(Z“"(1)) = E(EIZ“"(1) I F.]) = E(Z“(s)EIZ""<s;1) I F.1)= E(Z“(s)) = 1.

33

Observe that for each 11

(3-9) Ev"[Z(1) | El = E[Z(1)Zv"(8;1) | F5] = E[Z(1)Z”"(S;1) l Fsl =E""[Z(1) I F5],

so that Vo(s) = lian E""[Z(1) I F,]. Then that Vo(r) 2 E“[Vo(s) I F,] follows from

Vo(r) 2 lim supn E“"[Z(1) | F.] and (3.10); 2

(3.10) 13“"120) I 1:.) = E“"[E“" [Z(1)| F.] I F.] = BIE“"IZ(1) I F.] z“(r;s) I F.] and
min E[E”“[Z(1) | F5] z”(r;s) | F.] = E[v0(s)z“(r;s) | F.] = E“[vo(s) | F,] as.

So (3.6) holds, and V0 is a (P‘) supermartingale for each v e D.
The filtration {Ft} is right-continuous and complete. Therefore, to see that V0 admits
a modification with cadlag paths, it suffices to prove the equivalent condition that

t —> E(Vo(t)) is a right continuous function [See Lipster and Shiryayev (1977),
Theorem 3.1, page 55]. Fix t, and suppose tn > t, n = 1, 2, ..., satisfy t = limin t... For
each v in D, it holds for all n that Vo(tn) 2 E"[Z(1)I Ftn ] a.s. Also, for each v e D, since

(1 .20) implies that min05(51(Z"(1)') > 0 a.s., we have

E[Z(1)Zv(1)IF.l .

(3.11) ' E‘IZ<1)IF.1 =
2%)

 

Then the paths EV[Z(1)I F.] are continuous a.s., since E[Z(1)ZV(1)I F.] is a (P) martingale

with respect to the Brownian filtration {Ft}. Therefore, for each v e D,

 

2 In the second and final equalities in (3.10) we use the following consequence of that for each v e D,
ZV is a (P) martingale. If ‘t S B are stopping times and Y is an Fig-measurable random variable, then for any
v e D such that Y is (P') integrable,

“E“[YI F.] = ElY Z'(I;B)| F.].

34

lim infn Vo(tn) 2 EV[Z(1) I F t], implying lim infn Vo(tn) _>_ Vo(t) a.s. Then by the F atou

Lemma,

(3.12) lim inf,E(vo(t,)) 2 E(lim inano(tn)) 2 E(Vo(t)).

Since V0 is a (P) supermartingale by (i), E(V0(tn)) S E(Vo(t)) for each 11. Therefore
(3.12) implies that limnE(Vo(tn)) = E(Vo(t)). so we have the equivalent condition, and
the process denoted by V in (iii) exists.

For (iii), suppose \7 is an adapted process with (as) cadlag paths such that
{7(1) = 2(1) a.s. Suppose that there exists a q e Q = {rational q: 0 S q S 1} such that
P{ V(q) < V(q)} > 0. (By right continuity, { V(t) — V(t) 2 0 for all 0 S t S 1}c equals
I quQ{ V(q) — V(q) < 0} modulo null sets.) Apply Lermna 2 to obtain the monotone
convergence V(q) = lianEv"[Z(1) I Fq]. Then, for n sufficiently large, we see that \7

cannot be a (Pvn) supermartingale from equivalence of P and P"n and both {7(1) = 2(1)

(P) as. and P{ v(q) < EV“[Z(1) | Fq]} > 0. I

Corollary 1. P{V(t) S Z(t)f0r all 0 S t S 1} =1.

Proof. The result will follow from (iii) of Theorem 2 once we show that 2(a) is a (PV)

supermartingale for each v e D. For each v e D, any adapted process Y(-) is a (P‘)

supermartingale if and only if Y(-)Z"(-) is a (P) supermartingale. If v e D, we have
A®P{0(t,m) e Ran[o'(t,m)], v(t,a)) e (Ran[o‘(t,o)])i} =1,

so that Z(-)ZV(-) is (P) supermartingale 29*V(.). I

35

3.2 An Essential Lemma in the Link to the Arbitrage Problem.

Lemma 3 below will be essential in constructing an arbitrage in the proof of the final
theorem in Chapter 4. In the proof of Lemma 3, it is necessary to use the Doob~Meyer
Decomposition of a cadlag supermartingale. Below is the formulation of the Doob-

Meyer Decomposition Theorem given in KOpp (1984) [Theorem 3.8.10, page 122].

Theorem (Boob-Meyer Decomposition.) LetX be a right-continuous
supermartingale. Then X has a unique decomposition X = M - A, where M is a local

martingale and A is a predictable increasing process.

The conclusion of Theorem 2 that for each v e D, V is a (P‘) supermartingale with
cadlag paths implies that for each v e D we have the unique (PV) Doob-Meyer
Decomposition V = Lv - A". (Uniqueness is up to (PV) indistinguishability, which is the
same as (P) indistinguishability by equivalence of P and PV.) Each Lv is a (P‘) local
martingale, and each AV is an adapted process with cadlag and non-decreasing paths

satisfying A"(O) = 0.

Lemma 3. Let L denote the (P) local martingale in the (P) Doob-Meyer

decomposition of V. Then L(-) = V(0)Z°(o) for an adapted Rd-valued process (p satisfying

(3.13) A®P{<p(t,a)) e Ran[o'(t,a)]} = 1.

Proof. Because L is a nonnegative Brownian local martingale satisfying L(O) = V(O),
L(-) = V(0)Z“’(o) for an adapted Rd-valued (p satisfying 01‘p = 1 as. There exists such a (p

satisfying that if C(L) = inf {t > 0: L(t) = 0}, then for each (t, (1)), t 2 Q(L).,, implies

36

(p(t,o)) = 0. [See Lipster and Shiryayev (1977), Lemma 6.2, page 208.] (3.13) will

follow once we prove
(3.14) For each v e D, 71®P{(p’(t,a))v(t,m) S 0} = 1.

To show that (3.14) implies (3.13), suppose that (3. 13) does not hold. Then define the
process I10 by putting uo(t_.o)) the projection of (p(t,0.)) on Ker[0'(t,(t))] = (Ran[o"(t,a)])i.

Then we have that A®P{ uo(t,03) ¢ 0} > 0. Let process It be given by

110) = #1100). if 11.0) a o; 110) = 0 if too) = 0.
”110(1)“

The Novikov Criterion implies that us D because u is bounded uniformly in (t,c0).
(3.14) does not hold for this It, since (p'u = ”up”.
To prove (3.14), consider that for each v e D, where (PV) Brownian Motion W" is as

defined in (2.4), it is possible to write the (PV) Doob-Meyer Decomposition of V as

I II:
(3.15) V(t): V(O) + Irv (s)dw"(s) —AV(t), OStSl,
0

1
where f’ is an adapted Rd-valued process such that Illf” (t)II2dt < 00 as. With respect to
0

(3.15) note that for each v e D, Lv is adapted to {F1}, and not necessarily adapted the

P-augmentation of the natural filtration of W". If this latter filtration is denoted

{ Ftv ; 0 S t. S 1}, then it is possible that there exists an adapted v such that there exist t

such that FtV are strict subsets of the corresponding Ft. Therefore, it is not possible to

directly apply the Martingale Representation Theorem to obtain the stochastic integral

37

representation of L"(-) in (3.15). If V(O) = 0, then because V(-) is a nonnegative (P)
supermartingale, V(-) is indistinguishable from the zero process, and clearly we may take
F to be the zero vector process for each v e D. 3 If V(O) > 0, then for each fixed v e D,
to construct the process f’ start from that Lv(o) is a (PV) local martingale is equivalent to
that LV(-)Z"(o) is a (P) local martingale. Then since LV(.)Z"(.) is a nonnegative process,
there exists an adapted Rd-valued process g = g" satisfying org = 1 as. such that

L"(-)ZV(-) = V(0)Zg(.), Then the K0 formula gives that

dZ" (t)

(3.16) dD(t) = (1le = W» ng, V(0)Z‘(t)

2:0) 7(7) 9 ‘ (an)

 

— V(O) ),,a28(.)ZV(.) t) + V(O—)—-Zx(’)sz(),Z"(.) t)
(2%) [ I (2%)? [ I
V(O)Zg(t)

=_Z.V_()_Ig*(t)dwo) +v*(t)dW(t) —g*(t)v(t)dt +IIVII2(t)dt
I

= Lv(t)(v*(t) — g* (t)XdW(t) + v(t)dt)
So we have equation (3.15) with f’(-) = LV(-)[v(o) — g(o)].
Then because dL(t) = —L(t)(p(t)dW(t), 0 S t S 1, it follows from uniqueness of the (P)

Doob-Meyer Decomposition and equivalence of the (P‘) that for each v e D

(3.17) 7L®P{f’(t,0)) = —L(t,0))(p(t,(o)} = 1.

and consequently, up to indistinguishability,

 

3 If Y(t), 0 St S 1, a process with cadlag and non-negative paths. is an ((2, F, {F.}, P) supermartingale,
and S = inf{t > 0: Y(t) = 0}, then P{Y(t) = 0 for all S < t S 1} = 1. [See Elliott (1982), Theorem 4.16,
page 38.]

38

t
(3.18) A"(t)= A(t) — ]L(s)tp*(s)v(s)ds; OStSl.
0

Equation (3.18) motivates the completion of the argument. F ix v e D. Then define for

each n a process un e D, with E(Z""(1)) = 1 following from uniform boundedness of un

 

 

 

in (t,m):
Io“(t)v(t) > 0} Ia*(t)v(t) s OI
3.19 n t = t t .
Then, by (3.18),
1 at
(3.20) A”n (l) = A(l) — n I {‘8 (t)V(t) > 0jL(t)<p"‘(t)v(t)dt
0 1+llv(t>l|

_ ‘anmsoI .
OI 1+I|v(t)|I L(t)<p (t)v(t)dt

We must have A®P{(p'(t,a))v(t,co) S O} = 1 because for a) 6 {Mt (p’(t)v(t) > O}> 0}the
right-hand side of (3.20) tends to —oo as n —> oo. (Recall that for each (t,o)), L(t,o)) = 0

implies cp(t,ct)) = 0 6 Rd.) Such divergence would contradict that for each n the paths

A1L1 n(-) are non-decreasing.

39

3.3 Two Useful Characterizations of the Process V(o).

It will be useful that the following result holds when we stop process V. Karatzas and
Shreve (1998) give a different presentation of this proof which is based upon a similar

underlying approach [see Remark 5.6.7, page 215.]

Lemma 4. For each stopping time 0 S t S 1,
(3.21) V(t) = ess szrpveDKEv[Z(I) /F,]).

Proof. For each 1, denote the right-hand side of (3.21) by Y(t). First assume that 1: is

a simple stopping time, ‘t = 21515,, {t = tk}tk. Then for each v e D,

(3.22) EVIz(1)|F,] = §{e=tk}EV[Z(1)|F,k].
k=l

(The right-hand side of (3.22) is FT-measurable because {I = tk} O A e F I for any

A e Ftk , Equality of integrals on F: sets follows from that {t = tk} O A e Ftk for any
A e F.) Evident from (3.22) is that Y(T) 3 2,5,5,“ = amt.) = V(T).

For the reverse inequality, choose by Lemma 2 sequences {vk,m; m 2 1} c_: D
for k = l, ..., n such that for each k, limmIEvk’m[Z(1) IFtk ] = V(tk) a.s. Then, in light of

(3.22), it holds as. on {t = tk} that
Y(I) 2 liran Evkm[2(1) | F.] = V(t).

Now let I be an arbitrary stopping time. Stopping times Tk, k = 1, 2, ..., given by

. ‘-l . .
-J—on J——<rSi ,forj=l,...,k
k k k

40

satisfy ‘tk 2 t and lika/tk = t as. The paths EV[Z(1) I F.] are continuous for each v e D

and the paths V(o) are right-continuous, so that for each v e D it holds as. that
(3.23) 1332(1) | F.] =1imkEv[Z(1) 11:11.] s limkV(rk) = V(T).

Therefore, Y(t) S V(T) a.s.

Now take for each k _>_ 1 a sequence {vk,m; m 2 1} g; D such that
(3.24) V(tk) = luanE"“~'“[2(1) |I~‘Tk ] a.s.

Define adapted processes ukm, k 2 l, m 2 1, by “k,m(t) = vk,m(t){t > 11,}. Then each
um e D. (Calculation (3.3) shows that for any v in D and stopping time B, v(-){- > B} is
also in D.) For each k, we have (3.24) with the processes um replacing the processes

vbm. Furthermore, for each k,m, with the last equality in (3.25) holding because

ZN"m (1; H) = 1 as,

(3.25) Y(t) 2 E“"’"‘[Z(1) I F.1= E““’"‘[E“"“‘[Z(1) IF.,I I F.]
= E[E“"‘"‘[Z(1) Ink] | F.] as.

(Refer to footnote 2 on page 33 for more detail on this calculation.) Letting m —> 00 for
fixed k in (3.25) shows that for each k, Y(t) 2 E[ V(rk) I F.] as. Therefore, the Fatou

Lemma implies
(326) Bore» 211m sup. E(Emtl) I F.]) = 1m. E(V(rt)) 2 Ewe».

Then Y(t) S V(t) as. and (3.26) imply Y(t) = V(t) as.

41

Although properties of the processes Z"(o) dependent upon v e D were used in
proving properties of the process V(o) above, for each t, V(t) is in fact a version of an

essential supremum taken over a much wider class than D. Define the class N by
N = {adapted Rd-valued processes v: or" = 1 as. and l®P{v(t,m)e Ker[o(t,o))]} = 1}.
Lemma 5. For each stopping time 0 S t S 1,

(3.27) V(t) = ess supv.E)(r(E[Z(1)Zv (1:;1) /F.]).

Proof. D _c_ N implies "S". For "2", fix a stopping time t and v e N. Then define two

sequences, one of stopping times BH and the other of adapted processes v", by

t
(3.28) Bn = inf {t > O: I{t < s}IIv(s)II:2 ds 2 n},
0
(3.29) vn(t) = v(t){1: < t S Bn}.

Each vn e D. Clearly A®P{vn(t,m) e Ker[o(t,0))]} = 1, and we have for each n that

l
JIIv,(t)I|2dt S n as. The Novikov Criterion implies that E(ZV“(1)) = 1.
0

The paths Zv(r; .) are continuous a.s., and 2""(13 1) = ZV(T; B"). Therefore,

limn Zv"(t; 1) = Zv(r; C(13)) = Zv(‘t; 1) as. Conclude through the Fatou Lemma:
(3.30) V(t) 2 lim supn EV"[Z(1) | F.] = lim supn E[Z(l)Zvn(t; 1) | F.]

2 E[Z(1)ZV(1:; 1) | F.] as. I

42

Chapter 4

Characterizations of Arbitrage and Corollaries

4.1 Characterization of Arbitrage in Terms of Domination of Z(t; l).

The following simple fact is presented first to avoid repetitive justification.

Lemma 6. Let (p be an adapted Rd-valued process satisfying 01‘p = 1 a. s., and let
0 S t S 1 be a stopping time satisfiing P{r < I} > 0 such that processes Z‘”(o) and Z(t; .)
are not indistinguishable. Then there exists a constant 0 < 5 < 1 such that the stopping

time B defined by
B = inf{t > t : Z”(t) S 6Z(t; t) and Z(t; t) > 0}
satisfies P{B < I} > 0.

Proof. Suppose that no such 8 exists. Then since the paths Z‘p(o) and Z(t; .) are

nonnegative and continuous,
(4.1) P{Z"(t)2Z(t; t) forall‘tStSl} =1.

In particular, Z“’(1:) 2 Z(t; r) = l as. Then that Z‘”(-) is a continuous supermartingale with

Z"(O) = 1 as. implies

P{Z‘p(t) = Z(t; t) for all 0 S t S t} = 1.

43

Furthermore, (4.1) implies that that CWO) 2 §9(r) a.s. Then there exist localization
stopping times 1],, with limklnk = §°(r) as. such that for each k the process

Z"(- A m) — Z(r; . A m.) is a nonnegative martingale. Then 23(0) — Z(‘t; O) = 0 as.
implies that each of these localized martingales is indistinguishable from the zero

process. Then path continuity implies both that
(4.2) P{Z‘p(t) = Z(t; t) for all 0 s t _<_ 6(a)} = 1

and that Z‘°(§9(t)) = 0 as. on {§9(t) < 1}. Because Z‘”(-) is a nonnegative
supermartingale, we can replace §9(t) by 1 in (4.2). So Z‘p(-) and Z(‘L’; .) are

indistinguishable. I

The following theorem is an extension of Corollary 3 of Levental and Skorohod
(1995) [page 920]. It serves as the link between the V process detailed in the last section

and our arbitrage problem.

Theorem 3. Assume absence of immediate arbitrage. Arbitrage exists if and only if

there exists both an adapted Rd-valued process (p satisfying
A®P{cp(t,(1)) e Ran[o'(t,w)]} = 1 and P{a‘” = I} = I
and a stopping time 0 S ‘I.’ S 1 such that the processes Z‘p(t; .) and Z(t; .) are not
indistinguishable and P{Z‘” (1:; 1) 2 Z(t; 1)} = 1.
Proof. (Sufficiency) Lemma 6, applied to (p(-){- > 1}, implies that there exists a
constant 0 < 5 < 1 such that the stopping time B defined by

(3 = inf{t > a; 2““ (r; t) s 52(1; t) and Z(T; t) > 0}

satisfies P{B < l} > 0. Assume first that Z(r; 1)> 0 as. on {B < 1}. Then it also holds

as. on {B < 1} that

 

(,3) 1 2 2304) : [Zimmjvwnj . 52mm
Z(I;l) Z(nB) Z(B;l) Z(Bd)

So zt(8;1)/z(p;1) 2 8‘1 > 1 as. on {B < 1).

It follows from the Itc‘) Formula that on {B < t < §9(B)}

Zion) '_ Z‘Past)
(4.4) d{ 2033)] _ wt) ———(9* (t)— (p *(t)IdW(t)+9(t)dt).

(The computational details underlying (4.4) are precisely the same as those given
explicitily in calculation (3.16).) Let ((3'),. denote an adapted process such that for

each (t,w), (o')+(t,(o) is an invertible dxd matrix and

(4.5) o‘(t,to) (6);} (t, u) x = x for all x e Ran[o‘(t,m)].

Lemma 1 guarantees that such an (0'). exists. Then because of the range hypothesis on (p

and the construction of 0,

(4.6) l®P{o (on) (0— (p)= 0— (p}=1.

Choose the portfolio 71 as follows:

(47) n0) = Bari—LII," ;){B< <tl<o >130)(90)— 4.0))

We have assumed Z(‘t;l) > 0 as. on {B < 1}. So Z‘p(t;1)2Z(t;l)> 0 as. on {B < 1}.

Then since B 2 t,

45

(4.8) [{0 <1}Ip (1)132 +|Itp(t)|flit < 00 as.

The paths B(-){Z‘° (B; .) / Z (B; o)} are bounded in t as. because it holds as. that
nonnegative supermartingale Z(B; .) cannot hit 0 on {Z(t;l) > 0}. So, in light of (4.8), 7:

satisfies the integrability constraint required of a portfolio. Where random variable M is

given by
M = 5111305151 B(t)%(-IiB—;;t_t)2’
1 2
(4.9) IUIo*(t)n(t)“ + In*(t)a(t)|]dt s
o

(M2 +1) I11) «41910-00112 + I(e‘(t)-<p'<t)})(t)|ltt.
By construction,

' 'Z°<13;s) Z"(B;t)
4.10 X = d —— = _—
( ) .0) I [am] Z(B;!)

Observe that 7: is l-tame, and that Xn(l) = 0 as. on {B = 1}. Then that
z¢(p;1) /z(13;1) 2 8" > 1 as. on {(3 < 1}, where P{B < 1} > 0, completes justification of
that 1: is an arbitrage.

Now drop the assumption that Z(r; l) > 0 as. on {B < 1}, but treat the situation where
P({Z(T; 1) = O}O{B < 1}) > 0 under the added assumption that P{C"(B) 2 §6(B)} = 1.

The It?) Formula gives that on {B < t < Q9(B)},

46

 

 

1 1 .
(4.11) ((2033)) _ Z(B;t)9 (t)(dW(t)+9(t)dt)

With constant 5 and matrix-valued process (6’). as they were earlier in the proof, define

the portfolio rt' by

Z¢(B;’){

Z(B;!) 1) < r s v}(c*);‘(t)(e(t) — 40))

(4.12) rt'(t) = B(t)

K :1: -1
B(t)—__Z(13;t){B < r s vim )+ (090) ,

where K = (8"1 — 1) / 2 > 0, and st0pping time y is defined

7 = inf{t > 0: Z(B; t) = K /(2 + 2K)}.

Path-continuity on (B; .) and K /(2 + 2K) < 1 imply that a.s., Z (B; l) > 0 on {y = 1} and
(4.13) v < 49(1)) on {Z(B; 4903)) = 0} = I I{B < 911910124 = 00}.

Paths Z‘p (B; . A y) / Z(B; . A y) and K / Z(B; . A y) are bounded in t as. S0 inequality

(4.14) below shows that (4.13) and P{§¢(B) 2 §9(B)} = 1 together imply that n' satisfies

the portfolio integrability constraint. With random variable M' defined by

Z"(fl;t Ar)+ K
Z(fl;t)

 

M, = 3111305151 3(1)

9

47

(4.14) I{IIo*(t)a'(t)II2 + Ia'*(t)a(t)l}dt s
O
(M'2 + 1) I {p < t s y}I|o(t) — a(t)|I"' + |(e‘(t) — a‘(t))e(t)|}lt +

l
2(M'2 + 1) I {,6 < z s y}IIt9(t)H2dt.
0
(Since Z(t; B) > 0 on {B < 1} and Z(B; 1) > 0 as. on {y=1}, P{Z‘p (r; 1) 2 Z(r;1)} =1
1
implies that IIB < t}II(p(t)I|2 dt < 00 as. on {y = 1}. )

0

The associated discounted capital gain process satisfies

 

_ 2¢(/3s) K
(4.15) X20) - IMGQI d[Z(,B;s) + Z(fl;s)I

 

: Z"(B;t/\T) _ 1 + K
Z(B;t A T) Z(B;t A 1’)

So 7t' is (1 +K)-tame. On {B =1}, an(1) =0 a.s. On {B <1}fl{y< 1}, X..'(1)21+K a.s.

With regard to Xnv(l) on {B <1}fl{y =1}, recall that we have Z(T; B) > O on {B <1} and
Z(B; 1) > 0 as. on {y = 1}. Then calculation (4.3) shows that

28(13;1)/z(p; 1) 2 8“ as. on {B <1}n{y=1},

so that an(1) 2 6'1 — 1 — K > 0 as. on this set. So if is an arbitrage.

48

All that remains in the sufficiency argtunent is to produce an arbitrage given

P{C"(B) < §9(B)} > 0. In this case, define the portfolio it" by

B(t)

—————- I:“’(B) <t < p, 43(1)) < t9(B)}(o*);‘<t)e(t),
Z(C"(B);t)

(4.16) rt'(t) =

where stopping time p is defined

p = inf{t> 0: Z(Q“’(B); t) = 1 /2}.

If 48(9) < 49(0), then 116 «119101124 = 00, so that 2‘90); 1) = o as. on {43(0) < §°(B)}-

Since I S B, we also have Z“’(r; l) = O as. on this set. Therefore, Z‘p(t; l) 2 Z (t; 1) as.

implies

(4.17) z (r; 1) = 0 as. on {$10) < 49113)}.
On 13(1)) <ts 1).

(4.18) Z(T; t) = Z(T; B) Z(B; $903)) Z(C‘”(B); 1)-

We have both 20; B) > 0 on {9115) < 49(0)) c; {13 < 1). and Z(B; 43(0)) > 0 as. on
{Q‘p(B) < §6(B)}. Therefore, path continuity. (4.17), and (4.18) imply that p < §°(B) as.
on {§“’(B) < §°(B)}. Then a calculation like (4.14) shows that 7:" is a suitably integrable

portfolio. it" is an arbitrage since

1
Z(C“’(fl);t/\p)

 

(4-19) Xfl-(t) = I - 1]I§”(fl)<§9(fl)l

49

Observe that 1t" is l-tame, Xn~(1) = O as. on {C(13) _>_ 3(6)}, and Xn»(1) = 1 as on

{:m»<§%m)

(Necessity) Suppose that no is a C—tame arbitrage portfolio. Then defining
1: = (20’an produces an arbitrage for which P{Xn(t) + 1 > O for all 0 S t S 1} = 1.

Define the adapted vector process (p as follows:

1 .
(4.20) (0(1) _ a(z) - magma).

Then <p(t,o)) e Ran[o'(t,co)] for each (t,co). Because 1! is 1/2-tame and satisfies the

portfolio integrability constraint,

(4.21) ), “o'(t)7t(t)“2dt < 00 as.

1
l0+xgo

so that a‘” = OL = 1 as. (4.21) also implies that C” (1') = 6(1) a.s.

Now define the stopping time T by
r = inf{ t> O: Xn(t) ¢ 0}.

It follows from the definition of r and that dX,,(t) = B"(t)1t'(t)o(t)[dW(t) + 6(t)dt] that up

to a P-null set,

(4.22) {r<1}= {Masts1:o‘(t)n(t){rst<(r+g)A1}¢0) >0 foralls>0}.

P{a = 1} = 1 implies 6(1) > r as. on (I < 1}. That ‘11: is an arbitrage implies

P{r < 1} > 0. So (4.22) implies that

x®P(cp(t,m)(r,, < t < a(t),} i 6(t,w){rw < t < 6(1),») > 0.

50

Then to see that Z“’(r; .) and Z(r; .) are not indistinguishable, examine the explicit
formula for process Z"(1:; o), (1.11), and conclude that the map [v] —-) Z"('c; .) defined on

the equivalence classes of adapted Rd-valued processes v satisfying 01V = 1 as. partitioned

by the relation
V15 V2 if A®P(Vl(tam){‘tm < t < €Vl(T)w} = V2(t,(t)){‘tw < t < Cvz(1)w}) = 1
is an injective map.

Now calculate that on {O s t < §9(r)},

Z‘p(1:;t) _ ’ {t<s} a X
(4.23) ———Z(w) _ exp{ J——1+Xn(s)n (s)o(s)dW(s)

1 ' . 2 ‘ .
xeXp{-§6[(1—_EZX:S?})-)2-“G (s)n(s)|| ds + 01%n (s)o(s)6(s)ds}.

So Z‘p (I; .) / Z(r; .) is the unique strong solution on the stochastic interval

{(t,(o): 0 s t < gemw} to the SDE

(4.24) dF(t) = F(t)I:{_EY<-%Sn*(t)o(t)[dW(t)+9(t)a’t]; 1“(0)=1.

Since 1 + X..(.) solves (4.24), it follows that (1 + x..(.))(. < gem) and
(Z‘p(1:; .) / Z(r; -)){- < Q°(t)} are indistinguishable processes. Then reasoning on
{Z(‘r; 1) > 0} ; {gem = 1} that x, is an arbitrage capital gain, and on {Z(m) = 0} that

Z‘” (I; .) is a nonnegative process, we obtain that P{Z“’(t; 1) Z Z(‘L‘; 1)} = 1. I

51

Remark. Under the assumption that P{a = 1} = 1, the proof of necessity in the

previous theorem characterizes the capital gain process Xn(-) associated with an arbitrary

arbitrage 1t(-) as C({Z"°(r; .) / Z(r; o)} — 1), for a constant C > 0, on the stochastic

interval {(t,(t)): 0 S t < §9(r)m}. Here I = inf {t > O: Xn(t) at O}, and (p is an adapted process
satisfying a range requirement. Therefore, it is interesting to question if it is ever
necessary in producing arbitrage to hold asset shares at times t 2 gen) when §9(r) < 1,

i. e., at times t when the ratio Z“’('c; t) / Z(r; t) is undefined because Z(r; t) =10. The
answer is no. Assume absence of immediate arbitrage. If arbitrage exists, then there is
an arbitrage 1: such that, where T = inf {t > O: Xn(t) ¢ 0}, there exists a stopping time 7
such that y < 6(1) a.s. on {Z(t; 6(1)) = O} and X,‘(1) = Xn(y) a.s.

To justify the assertion in the preceding paragraph, suppose it is an arbitrage. Put
stopping time i = inf {t > O: X {I (t) :t 0}. By the proof of necessity in Theorem 3, there
exists an adapted Rd-valued (p satisfying a” = 1 a.s., A®P{<p(t,co) e Ran[o'(t,(o)] } = 1,
Z“’( "r ; .) is not indistinguishable from Z( ’c ; o), and P{Z‘”(i:; 1) 2 Z( i: ; 1)} = 1. The
proof demonstrated that this (1) satisfies P{Q‘”( ’c) = Q9( % )} = 1. Therefore, we can define

an arbitrage 1t using the same construction as for the arbitrage n' in the proof of

sufficiency in Theorem 3. Define
(4.12) to) = B(t)%‘%%?p <1: 11(6):} (t)(e(t) — 4(1))

K
Z (13;!)

 

81:) {1) <1 s 1}(o*):‘<r)e(r).

Here, there exists a O < 8 < 1 such that

52

B=inf{t> %:Z‘P(i ;t)SBZ(%;t)andZ(% ;t)>0}
satisfiesP{B<1}>O, K=(5"—1)/2 > o, and

y = inf{t > B: Z(B; t) = K /(2 + 2K)}.

1: is an arbitrage, and Xn(1) = Xn(y) a.s. Let T = inf{t > 0: X,,(t) at 0}. Note that

i <5 s r s 7 holds a.s. on {i < 1}. We have 7 < cam) a.s. on {Z(B; 6(5)) = 0}

[c. f. (4.13)]. Therefore, 13 s 1: < 6(1)) a.s. on {Z(B; @903» = 0}, and consequently,

4%) = @945) a.s. on {Z(B; c911)» = 0) and {211; do» = 0} = {Z(B; 4113)) = 0} a.s. So

7 < 6(1) a.s. on {Z(r; 33(1)) = O}, and we have proven the assertions in the remark.

4.2 Characterization of Arbitrage in Terms of the Processes V(r; .).

Lemma 7. Assume P{a = 1} = 1. Then Z(r; o) is indistinguishable fi-om V(r; .) for
all constants 0 _< r S 1 If and only if Z( r; .) is indistinguishable fiom V(r; .) for all

stopping times 0 S 2'5 1 .

Proof. Sufficiency is obvious.
(N ecessity.) First prove that for any stopping time T, in order to prove Z(‘t; .) and

V(r; .) are indistinguishable it suffices to show
(4.25) For each 0 S t S 1, V(r; t v r) 2 Z(‘L‘; t v t) a.s.

Corollary 1 states that P{V(T; t) S Z(‘L’; t) for all 0 S t S 1} = 1, so that we may replace "_>_"
in (4.25) by "=". The paths V(r; .) are right-continuous, and the paths Z(‘t; .) are

continuous. So (4.25) implies

53

P{V(‘C;t)=Z(T;t) foralltStS l} = 1.

In particular, V(T;‘r) = Z(t;r) = 1 a.s. Since V(r; .) is a (P) supermartingale by Theorem 2

and V(r; 0) S Z(‘C; O) = 1 a.s., we then have
For each 0 St S 1, V(r; tA'c) = 1: Z(T; t /\ T) a.s.
Then by right-continuity, Z(‘C; .) and V(r; .) are indistinguishable given (4.25).
Now let 1' be a simple stopping time, T = Zfir = tj}tj. Fix 0 S t S 1. Because of
indistinguishability of Z(tj; .) and V(tj; .) for each j, Lemmas 2 and 4 imply that for each j

there exists a sequence {V1.16 k 2 1} g D such that a.s.,

(4.26) lika EV” [Z(t j;1) Fm] = V(tj; t v r) = Z(tj; t v r).

 

Since for each v e D,

 

 

(4.27) E"[Z(t;1) FM] = 2,-{r = tj } EV[Z(t j;1) PW]
it follows that
(4.28) lika E"1-*[Z(r;1)l=,,,] = Z(‘C; t v 1:) a.s. on {r = tj}.

 

Since supJ-,kEV"k [Z(‘C;1)Ftvt] S V(r; t v T) a.s., (4.25) holds for this simple I .

 

Now let 0 S 1: S 1 be any stopping time. There exist simple stopping times
In 2 1' with limnlrtn = r a.s. Fix 0 S t S 1. By the preceding paragraph, for each n
V(t v In; .) and Z(t V In; .) are indistinguishable processes. So for each 11, there exists a

sequence {vm k; k 2 1} g D such that a.s.,

54

(4.29) lika Ev” [Z(t v Tn;l) Ftwn] = V(t v In; t v 1,.) = Z(t v tn; t v In) = 1.

 

For each n, k, the adapted process link, defined by pn,k(s) = vn‘k(s){s > t v In} lies in D

[see (3.3)], and we have

Ew‘ [Z(t v 7:51le, = E“ [Z(t v 7,51) EV," ].

 

 

For each n, with the last equality below holding because Z"""‘ (t v r;t v In ) = 1 a.s. for

each n, k [see footnote 2, page 33],

(4.30) V(t;tv1) 2 likaE“"~*[Z(r;l)F,,,] = Z(r;tvr) lirnkTE““"‘[Z(tv1:;1)FM]

 

 

= Z(T; t v ‘1') lika E ”""‘ [Z (t v r;t v 2'")E”"'* [Z(t v r";1) EWJEVJ

 

= Z(‘t; t v T) lika E[Z(t v 1:;t v tn)E"“'k [Z(t v In ;1) PW," 117nm].

 

Then monotone convergence (4.29) yields that for each n,
(4.31) V(t; t v 1') _>. Z(‘C; t v r) E[Z(t v t;t v en)|F,,,].
We have lirnn Z(t v 1.“; t v In) = 1 a.s., so that (4.31) and the Fatou Lemma imply

V(‘t; t v 1:) 2 Z(‘L'; t v I) a.s. I

Theorem 4. Assume absence of immediate arbitrage. There is no arbitrage if and

only if processes Z(r; .) and V(r; .) are indistinguishable for all constant times 0 _< r S1.

Proof. (Necessity) Suppose that Z(r; .) and V(r; .) are not indistinguishable.
Consider the (P) Doob-Meyer Decomposition, cadlag V(r; .) = L(-) — A(-). Lemma 3

implies that L(o) = V(r; O)Z"’(o) for an adapted Rd-valued process (p satisfying ()t‘p = 1 a.s.

55

and X®P{cp(t,o)) e Ran[o‘(t,m)]} == 1. A(-) is an adapted process with non-decreasing

and cadlag paths starting at A(O) = O. From

V(r; O) S Z(r; O) = 1 a.s. and V(r; 1) = V(r; O)Z“’(1) — A(1) = Z(r; 1) a.s.,

we obtain that Z‘”(1) 2 Z(r; 1) a.s. It cannot be that Z“’(-) and Z(r; .) are indistinguishable:
otherwise, in order for V(r; 1) = Z(r; 1) a.s. to hold, we would need both V(r; O) = 1 and
A(1) = O a.s. These values would contradict that Z(r; .) and V(r; .) are not
indistinguishable because the paths A(-) are non-decreasing. By Lemma 6 then, there

exists a stopping time B 2 t for which

(4.32) P{p<1}>0 and O<Z‘°(B)<Z(r; 13) on {13<1}.

That Z‘”(1) 2 Z(r; l) as and (4.32) imply Z“’(B; 1)_>_ Z(B; 1) a.s. If P{Z‘p(B; 1) > Z(B; 1)} > 0,
then Z‘p(B; .) and Z(B; .) are not indistinguishable, so that Theorem 3 implies that an
arbitrage exists. If Z‘”(B; l) = Z(B; 1) a.s., then Z‘°(1) 2 Z(r; 1) a.s. and (4.32) imply that

Z(B; 1) = O a.s. on {[3 < 1}. In this case, apply Theorem 3 with o, the zero vector process,

in the role of (p. We have o(t,(n) e Ran[o‘(t,co)] for each (Leo) and
(4.33) 2°43; 1)=1 2{B=1}=Z(B;1).

Z°(B; .) and Z(B; .) are not indistinguishable, and so an arbitrage exists.

(Sufficiency) Suppose that there exists an arbitrage given a = 1 a.s. By Theorem 3,

there exist both an adapted Rd-valued process (p satisfying or‘p = 1 a.s. and

X®P {<p(t,w) e Ran[o’(t,a))]} = 1 and a stopping time 0 S t S 1 such that the processes

56

Z“’(r; .) and Z(T; .) are not indistinguishable and Z“’(t; 1) 2 Z(‘L’; 1) a.s. Using these two
processes, we will construct an adapted process V(-) with continuous paths which is a

(P) supermartingale for each v e D, which satisfies 7(1) = Z(t; 1) a.s., and such that
there exists a stopping time y for which P{ V(y) < Z(t; y)} > O. The existence of such a

V(-) implies through (iii) of Theorem 2 that Z(t; .) and V(‘t; .) are not indistinguishable.
Then Lemma 7 implies that there exists a constant time r such that Z(r; .) and V(r; .) are
not indistinguishable.

Lemma 6 implies that there exists a stopping time B > T such that
(4.32)’ P{B<1}>Oand0<Z“’(t; B)<Z(‘C; B)on{B<1}.

Let g:[O, 00) —> [0, 1] be a continuous and strictly decreasing deterministic function

satisfying g(O) = 1. Define stopping times 7 and n as follows:
(434) Y = inf{t > BI Z‘p(T; t) = g(t — B)Z(t; 0}
n = inf{t > y: Z“°(t; t) = Z(t; t)}.

Path-continuity of Z‘p(t; .) and Z(‘t; o), (4.32)’, and that Z‘p(t; 1) 2 Z(‘L’; 1) a.s. together

imply that a.s. on {p < 1}

(4.35) Z“’(t; 7) = g(11 - B)Z(t; )1) < Z(t; Y) and Z‘”(t; n) = Z(t; It).
Now define 17(.)by

(4.36) V(t) = a((lt A 11 - B)”)Z“<t),

where the adapted process p(.) is given by

57

(4-37) 110)": 9(011' < t 3 Y} + (PONY < t S 71} + 9(017] < t}-

ThenforOStSl,

(4.38) to) = g((1t A y] - 1))*)Z<r; t A y) Z°’(Y; t A n) 2m; t).
so that in light of (4.35),
(4.39) yo) = so — B)Z(t; y) = 2%; Y) < Z(t; 11) as on {1) < 1)

and V(l) = Z(‘t; 1) a.s.

All that remains to accomplish is to show that V(-) is a (P‘) supermartingale for each

v e D. Since u(t, co) 6 {B(t, (D), tp(t, (13)} for each (t, to), we have

i®P{p(t,to) e Ran[c‘(t,to)]} =1.

As in the proof of Corollary 1, this fact and that Ran[o'(t,o))] = (Ker[o(t,(1))])i for each
(L(t)) imply that Z“(-) is a (P‘) supermartingale for each v e D. We are done, because

V(-) is the product of Z“(-) and an adapted process with non-increasing paths. I

Theorem 4 characterizes absence of arbitrage in terms of the processes V(r; o), which
are, loosely speaking, stochastic suprema. The following theorem improves the
characterization by expressing the result in terms of maxima at which these suprema are

attained.

Theorem 5. Assume absence of immediate arbitrage. There is no arbitrage if and

only if for each constant time 0 S r S 1 there exists a u e D such that E(Ze+“(r; 1)) = 1.

58

Proof. (Sufficiency) As justified in the proof of Lemma 7, for each I, to show that

V(r; .) and Z(r; .) are indistinguishable it suffices to show that

(4.40) For each t such that r S t S 1, V(r; t) = Z(r; t) a.s.

Fix 0 S r S 1. There exists a u e D such that the process 29+“(r; .) is a martingale. Then
HrStSl,

(4.41) E“[Z(r; 1) 1 F.] = EV[z°*"(r; 1) 1 F.] /Z“(r; t) = Z(r; t).

Therefore V(r; t) = Z(r; t) a.s. It follows that V(r; .) and Z(r; .) are indistinguishable for
each 0 S r S 1, and so Theorem 4 implies that there is no arbitrage.

(Necessity) Fix 0 S r S 1. Theorem 4 implies that V(r; .) and Z(r; .) are
indistinguishable. Let {an ; n 2 1} be a strictly decreasing sequence in (O, 1) such that
limnylr 8,, = 0. Since

V(r; r) = Z(r; r) = 1 a.s.
and

V(r ;r) = ess sup ..o E[Z°+"(r; 1) 1 F.].

by taking 1.11 = v9) for n sufficiently large, where {ng ; n 2 1} is a sequence in D such

(1)
that limnT E[ZG+V" (r;l)| F.] = V(r; r) = 1 a.s., we obtain that there exists a 111 e D such

that
E(z9+“1(r;1)) > 1 — c1.

Define a sequence of stopping times {[31,}. ; k 2 1} by

51.1: = inf{t>r: Ze+p1(r;t) v Z“1(r;t) =k}

59

Then 81,1. = 1 for k sufficiently large holds as., so that the monotone convergence

theorem implies that there exists a k; such that

(4.42) E(29+“1<r;1){1h,t, =11) A E(Z”‘(r;1){Bl,k1 =11) > 1 — a.

(111 e D implies that Z”1 (.) is a martingale process.) Denote 1:1 = 131, kl .
Now consider that the absence of arbitrage, Theorem 4 and Lemma 7 together imply

that V(‘L’l; .) and Z(t]; .) are indistinguishable. So there exists a sequence (v?) ; n 2 1}

 

in D such that

(4.43) limnT 13129"le(1:51))?Tl ] = 1 a.s.
Then

(4.44) 1im.E(ZG*“I maze”? (151)) =

(2)
limnTE(Ze+“1(r;tl)E[Z9+V" (t1;1)F,l]) = E(Ze+“1(r;rl)).

 

Because 29+“) (r;o /\ TI) is a uniformly bounded local martingale, and therefore a

martingale, E( Z 6+” (r;t])) = 1. Therefore, there exists a 112 e D such that

E(Z9+“1 (r;tl)29+“2 (t1;1)) > 1 — s2.

By considering the stopping times (132,1. ; k _>. k; + 1} defined by

9
Br... = inf{t>‘tt : ze+“l(rn.)z Wont) v z“1(r;tt)z“2<tr;z) =k)

6O

and monotone convergence, we obtain a st0pping time 1:; Z I] such that the processes
ZGWI . 9"“2 . 11 . 11 . - -
(r, . At1 )Z (11 . . A12) and Z l (r, . Atl)Z 2 (11, . A12) are umformly mtegrable
martingales, and
(4.45) E( 29“”1 (r; r,)ze"‘12 (t1; 1){r2 =1}) A E( z“1 (r; r1)z“2 (11; 1){r2' = 1} )
> 1 — 82.

Then by induction, there exist a sequence of processes {un ; n 2 1} in D and a sequence
of stopping times to S 121 S 12 S (put to = r) such that the processes Gn(-) and Hn(o)

defined by

' n' 6 . n _
(4.46) 6.1-) = 1’12 +“j(Tj—1;'/\Tj)s H.(-) = HZ“J(t,--i;-Ar,-)
. j=l j=1

are uniformly bounded martingales satisfying

(4-47) E(Gn(1){tn =1}) AE(Hn(1){Tn =1}) > 1-8n.

Define the adapted Rd-valued process 1.1 by

(4.48) 11(1) = 2141mm «5 tn}.

n=l

Then A®P{ 110,19) 6 Ker[0'(t,ot))} = l. on“ = l a.s. so that the processes Z“(-) and 29+“(.)

are well defined and continuous. Our method of selecting the stopping times tn imply

that there exist integers N1 < N2 < such that for each n,
r, = inf{t > o : zero; t) v Z“(r; t) = N}.

61

Therefore P(Un{‘tn = 1}) = 1. So the summation in (4.48) has finitely many non-zero

I
.,
summands as, and consequently, fl|p(t)“"dt < 00 a.s. By monotone convergence,

0

E(Z“(r;1)) = limnT 15(H,.(1){tn = 1}) = 1.
So 1.1 e D. Similarly, to complete the proof,

E(Z°*“<r;1)) = lirnnt E(G,(1){r, =11) =1.

The remark following the proof of Theorem 3 implies that in the absence of
immediate arbitrage, existence of arbitrage implies that there exists an arbitrage 1t
satisfying that for each to, 1t(t,(o) = 0 for all t 2 3(1),”, where stopping time t is defined by
1.’ = inf {t > 0: Xn(t) ¢ 0}. The folloWing result is interesting in comparison with that
conclusion. Note that the assumption of Corollary 2 below implies that for all stopping
times 0 S ‘t S 1, 6(1) = 1 a.s. Loosely speaking, we need check for indistinguishability of
V(r;o) and Z(r;o) for r > 0 only if an explosion of 9 occurs before time 1 and "kills the

market initiated at time 0.”

Corollary 2. Assume P{Qem) = I} = I. Then there is no arbitrage if and only if

processes V(-) and Z(-) are indistinguishable. Equivalently, there is no arbitrage if and

only if there exists a 11(o) e D such that E(Z9 +11(I )) = 1.

Proof. The proof of Theorem 5 establishes the equivalence of the two

characterizations. For necessity, apply Theorem 4 with r = O.

62

(Sufficiency) Assume V(.) and Z(-) are indistinguishable. Since P{gem) = 1} = 1
implies P{a = l} = 1, Theorem 4 applies. Since Z(l; .) is indistinguishable from the

constant 1 process, V(l; .) is indistinguishable from Z(l; o). Now fix 0 S r < 1. As in the

proof of Lemma 7, to prove that V(r;-) and Z(r;o) are indistinguishable it suffices to show
(4.25)’ For each t with r S t S 1, V(r; t) 2 Z(r; t) a.s.

If r S t S 1, then by Lemma 2 there exists a sequence vn in D such that

(4.49) limnTEV"12(1)1F.] = V(t) = Z(t) a.s.

Equivalently, since r S t,

(4.50) Z(t) = Z(r)(lim,,TEV"[Z(r; l)11=.] ) a.s.

P{§9(O) = l} = 1 and r < 1 imply that Z(r) > O a.s. so that a.s.

(4.51) V(r; t) 2 limnT Ev"[Z(r; l)|Ft] = Z(t) / Z(r) = Z(r; t).

In the statements of the remaining corollaries, recall that we assume absence of
immediate arbitrage, i. e., we assume a = l a.s. The following result is generally
understood; Levental and Skorohod (1995) prove it without reference to invertibility of 0'

[see page 917]. Here it is an easy corollary.

Corollary 3. If E(Z(t; 1)) = I for all stopping times 0 S r S I, then no arbitrage

exists.

63

Proof. The zero vector process lies in D. Apply Theorem 5. I

The following Corollary explicitly formulates the link between Theorem 5 and the

notion of an equivalent local martingale measure for the stock price processes.

1 .
Corollary 4. Assume j H9(t)”’dt < oo a.s. Then there is no arbitrage if and only if for
0

eachi= 1, d,

6151(1) = 5,0) Brandi/1.0),
lSde

where there exists a probability measure Q ~ P such that W(-) is a d-dimensional

standard Brownian Motion with respect to (Q, F, {Ft}o 5 t S 1, Q).

Proof. No arbitrage is equivalent to that there exists a v(-) e D such that

1~:(z9 * v(1)) = 1. Let W.) be given by
w, (t) = Wi (t) + 1(e,(s) + v,(s))ds; 1 s i s d.
0
Apply Girsanov's Theorem with probability measure Q equivalent to P defined by

dQ/dP =z"*V(1) . I
l 7
Under the assumptions Hp (t)||‘ dt < 00 a.s., Z(l) is a.s. bounded, and E(Z(1)) < 1,
0

Levental and Skorohod construct an arbitrage which does not require invertible volatility
[(1995), Example 5, page 924]. The following corollary accomplishes the same result

under a slightly weaker integrability assumption.

64

Corollary 5. If 6(0) = 1 a. s. and there exists a constant C for which Z(I) S C a. s.,

then there is no arbitrage if and only ifE(Z(])) = 1.

Proof. (Sufficiency) If E(Z(1)) = 1, then the proof of Corollary 3 implies that Z(-)
and V(-) are indistinguishable. Then Corollary 2 implies absence of arbitrage.

(Necessity) For each v e D and O S t S 1, EV[ Z(1)| Ft] S C. Then it follows from
right-continuity of paths V(o) that P{V(t) S C for all 0 S t S 1} = 1. Theorem 4 states that
absence of arbitrage implies that Z(-) and V(-) are indistinguishable. Therefore, Z(-) is a
uniformly bounded local martingale. (3(0) = l a.s. implies that we can take 1 as the
limit of the localization stopping times.) A uniformly bounded local martingale is a
martingale. [See Revuz and Yor (1991), Proposition IV.1.7, page 118.] Therefore,

E(Z(1))= 1. I

65

Chapter 5

Examples

In all examples, assume that r is the zero process.

Example 1. This is an example where E(Z(1)) < l and Z(l) > O a.s., but V(-) and
2(0) are indistinguishable. This example shows through Corollary 2 that when 0(a) can
be singular, the result that E(Z(l )) < 1 implies existence of arbitrage is not true. The

example can be written in one dimension, but is easier to read in a construction with

d=2.La

l 0
(Y(t) E [O 0].

To define 9, let 0 = to < t; < t; < with limnTtn = 1. Where W = (W1, Wz)‘, define for

k 2 1 sigma algebra
G. = a(wza) — W2(tk-1); tk_1 S t < tk).

Take for k 2 1, AR from Gk satisfying both P(Ak) < 1 and P(flkAk) = 1/2. Define a
deterministic process f(o) satisfying af= 1 a.s. by f(t) = ((1 — t)", O) ', O S t < 1, and a

stopping time t by ‘L' = inf{t > t]: Zf(t1; t) = 1/2}. Then define 0(a) by

66

(0,0)' for tStl
(5.1) 9(1): [hAjfl{th}]f(z) for tk<tStkm k21.

9 is adapted since each A. e ka . Note that for each (t,00), 9(t,c0) e Ran[o'(t,00)] as is

1
consistent with the general construction of 9. Since fl] f (s)“2ds = 00 a.s., P{r < 1} = 1.

'1

Then 6(0) and b(-) = o(o)9(o) = 9(0) satisfy the market parameters integrability constraint:

1 1
1
(5.2) 4201143) + ;le-(sflds S 1 + 6]{t1<sSt}i—ds < 00 a.s.

. -S
0 Mt

That I < 1 a.s. and that C (t1) = l for each 0) imply that Z(l) > 0 a.s.

To see that E(Z(1)) < 1, consider separately integrals over the cells of the partition

Q: (flAk)U(UBk), where Bk: A§\(UA°.) for k21.
k_>_1 k21 j<k J
Wehave

(5.3) 2(1) = Zf(t1;1:)= 1/2 a.s. on flkAk.

Furthermore, it follows by induction on k that1

(5.4) E(Z(l); B1) = P(Bk) for k 2 1.

(5.4) holds at k = 1 since B1 = Ac so that on B1, Z(l) = Zf(t1; t1) = 1. Now suppose that
. 1 |

(5.4) holds for j = 1, k—l. Note that, with F3 denoting the P-augmentation of

 

' For this example denote for Y integrable and A e F, E (Y ; A) = JYdP.
A

67

C{Wj(5), O S s S t} for j = 1, 2, Zf(t1; . A t) is { F,l }-adapted. Calculate, using in the third
equality that the F,1 V Fig} - measurable random variable Zf(t1; tk A r){flj<kAj-} is
independent of Gk :

(5.5.a) E(Z(l); Bk) = E(Z‘(ti; 11 A t) ; AE Wit-«Ar»

= E(E[Zf(t,; tk A T){nj<kAj}| Gk]; Air)

= P( A: ) E(Zf(t1; t1; A chm-(RAJ).

Through uniform boundedness in (gm) of f(-){- S tk}, Zf(t1; . A tk A I) is a martingale
process. Continue (5.5), reasoning in the second equality below that B(t) = O for all t > tk

on UJ-(kBj.
(5.5.b) E(Z(l); B1) = P( A: ){1 — E(Zf(t1; tk A t); (ma/3.93}

= P(Afi ){1 — EjakE(Z(l); 31)} = P(Afi ){1 - 2i<kP(Bj)}

=.P(A:)P(r),-aA,-) = P(Bk).

Combining (5.3) and (5.4) yields that E(Z(1)) = 3/4 < 1.

It remains to show that V(-)and Z(-) are indistinguishable. Since
P{V(t) S Z(t), for all 0 S t S 1} = 1,

continuity of Z(-) and Lemma 5 imply that it suffices to produce for each fixed 0 S t S l a

(1) e N for which Z(t) = E[Z(1)Z“’(t; 1)! Ft]. To this end, choose k such that tk-1 > t, and

68

consider the Gk- measurable random variable Y = Afi / P( Afi ), which satisfies E(Y) = 1.
There exists an adapted Rl-valued process (1)0 satisfying 01‘p0 = 1 a.s. such that if the
Rz-valued process (p is defined by (1)(t) = (O, (po(t))', then the processes E[YI 17.2] and
Z“’(-) are indistinguishable. (p e N, since the definition of 6 results in that

N = {adapted v = (v1, v2)‘ : onv =1 a.s., A®P{v1(t,m) = 0}=1}.

To prove Z(t) = E[Z(1)Z“’(t; 1) I F1], show that Z(-)Z‘p(t; .) is a (P) martingale. Since Y is

Gk-measurable with pH > t, and therefore independent of F3 ,

 

X®P{s St implies (p(s,(o) = O} = 1.

So Z‘p(t; .) and Z‘p(-) are indistinguishable. In particular Z‘p(t; 1) = Y a.s. For any v e N,
Z(o)Z"(-) is the (P) supermartingale 29%). Then that 2(.)z¢(t; .) is a (P) martingale

follows from that

(5.6) E(Z(I)Z‘°<t;1))= P( A: )“E(Z<1); At) = 1.

To justify the final equality in (5.6), consider the disjoint union

A: ,= UlAanjl

j=1

Ifj < k, then
(5.7) E(Z(l);A1°( m Bj) = E(E[Z(tj)A§ n leth D

= P(Ai)E(Z(1);B,~) = P(Ai)P(B,-) = P(Aiij).

69

In the first and second equalities in (5.7) we use that on Bj, Z(l) = Z(tj) a.s. For the

second equality also reason that if j < k, Alf e Gk is independent of Ft, . If j = k, observe
J
that Bk 0 AE = Bk , so that (5.4) implies
E(Z(l); Afi r) B1) = P(Afi 1) B1).

Example 2. By demonstrating a case where V(O) = l a.s. and V(-) is not
indistinguishable from Z(o), this example shows that we cannot reduce the conditions
equivalent to the absence of arbitrage to the behavior of the processes V(r; .) at time 0.
Note that the example exists under the restriction that Z(l) > 0 a.s.

Let d = 1, and a(t) = {t >_ 1/2}, 0 S t S 1. Let deterministic f(t) = (l—t)"l, for

O S t < 1, and define a stopping time t by
t = inf{t > in; z‘(1/2, t) = 1/2}.

Choose A from Fla with P(A) = 1/2; then define 6 as follows:

0 (t) '= f(t)(A n {1/2 < t st })
Take b = 09 = 9. This model is valid. 9’ is adapted, because A e Fm. 9 e Ran[o"] for
all (t,00). Moreover, P{‘C S l} = 1, so that 0' and b satisfy (1.4), the market parameter
integrability constraint. P {t < 1} = 1 also implies that Z(l) = Ac + 1/2 A a.s.
That Z(-) and V(-) arenot indistinguishable follows from F1 )2 measurability of Z(l ).
We have V(1/2) = 2(1) a.s., and Z(1)< Z(1/2) a.s. on A. To see that V(O) = 1, consider

the nonnegative {Pd-martingale Y(o), with Y(t) = E[2A°I Ft]. Since Y(O) = E(2A°) = 1,

7O

Y(-) = Z”(-) for an adapted process (1) satisfying d” = 1 a.s. (Take (p(t) = O for t > Q‘p(0)
for each 0).) Because Ac 6 F 1/2,

A®P{tp(t,0)) = 0 for all t > 1/2} = l, i.e., 7t®P{(p(t,00) e Ker[o(t,0))]} = 1.
So (1) e N. Then by Lemma 5,

V(0) 2 E(Z(1)Z“’(1)) = E(2Ac) =1.

Example 3. This example shows that we do not have in general that V(-) is a
continuous process. Here P{V(1/2) at lim ma V(t)} > 0. Let d = 1, and let
0 = to <11 < with limn tn = 1/2. Define Gk = c(W(t) - W(tk-1),tk_1 < t S t.) for each
k 2 1. Choose a sequence of sets Ak, such that Ag 6 Gk and P(Ak) < 1 for each k, and
P(flkAk) = 1/2. Define A = flkAk 6 F1 /2, and let 6(0), 9(-) and b(-) be as in Example 2,
using this particular A when defining 6(a).

Fix 0 S t < 1/2. Take k so that tk_1 > t and define a martingale process Y(o) by
Y(t) = P( Afi )‘1E[Afi I Ft]. As argued for similar processes in earlier examples,
Y(-) = Z‘p(o) for an adapted process (1) satisfying 06" = 1 a.s. and such that (p(t) = O for
t > §°(O) holds for each 0). Ak e Gk c Fm. So 7L®P{s > 1/2 implies (p(s,o)) = 0} = l;
equivalently, (p e N. Ag 6 Gk also implies through independence of G. and Flt—1 that
A®P{s < tk-1 implies (p(s,00) = O}= 1. Then t < lg-) implies Z‘p(1) = Z“’(t; l) a.s. Now,

using in the second equality below that Afi ; AC , where Z(l) = 1 a.s. on Ac, and using

independence in the final equality,

71

(5.8) V(t) 2 E[Z(1)Z‘°(t;1)lFt] = 1512(1))111)! F11=P<Ai)"E1A.‘.3 I F.]

= P(A: )-‘1>(A:).

Since I < 1/2 implies Z(t) = l as, and V(t) S Z(t) a.s., we have V(t) = 1 a.s. for each

t < 1/2. So lirn ma V(t) = l a.s. From Example 2, we know that
V(1/2) = 2(1) = Ac +1/2 A a.s.
Then P{V(1/2) at limm/z V(t)} = 1/2.
Example 4. This example is the central object of Delbaen and Schachermayer
(1998b). The Z process of this example satisfies EZ(1) < 1 and Z( 1) > 0 a.s.

Furthermore, there exists a v e D such that Z(-)Z"(-) is a martingale. Given such Z and

v,foreachOStSl

E[Z(1)|E]= 53519121020115] =20).

so that the processes V(o), Z(o), and E"[Z(1) I F.] are mutually indistinguishable. Because
Z(1)> 0 a.s., Corollary 2 implies absence of arbitrage.
One can construct a similar example in a market with d = 1, but the expression below

using (1 = 2 is nice in terms of readability. Define

1 0
0(1) a [0 0}

so that for all (t,(D), Ker[o] = {x 5 R2: x1 = 0}. Define the deterministic processes f and

gfortimesOSt<lby

72

f(t) = [l/(l—t), 0]', g(t) = [0, l/(I—t)]‘.
Define two stopping times 1' and B by
T=inf{t>0:Zf(t)=l/2}, B=inf{t>0:Zg(t)=2}.
Since Zf(1) = O a.s., we have t < 1 a.s. Define the process Z by stopping Zf: put
Z(t) = Zf(‘t A 11 At).
So defining Z is equivalent to defining process 9 by
9(t) = [(1/(1—t)){ts t A B}, 01‘.

Observe that 9 satisfies 60,00) 6 Ran[o’(t,o))] for all (L(t)). If b = 66, then b = 9, and the

market parameters integrability constraint (1.4) holds because 1' < 1 as:

1 1
#201,120) + Z|br(t)!}dt =1 + I{t$1/\B}Il—tdt.

0 LI 1' 0 —
Check first that that E(Z(1)) < 1:

(5.9) E(Zf(t/\B)) = 12f (t)dP + jzf(rA13)dP.
{B=1} {13<1}

P{B = l} = 1/2 because bounded local martingale Zg(B A t) is a martingale with Zg(B) = 2

on {1} < 1} and Zg(B) = 0 a.s. on {13 = 1}. Therefore, the first summand in (5.9) is

1/2 P{B = 1} = 1/4. To calculate the integral over {B < 1}, use independence of B and the

process Zf(- At) in the first equality below, and in the second, that for any t e [0, 1),

{Zf(S/\T), F5; OS 5 S t} is a martingale by boundedness of f(-){- S t}.

73

(5.10) jz/(ti At)dP = [E(Z/(tAt))P{13edz} = P{B<1}=1/2.
{13d} 10.!)

So E(Z(1))=1/4 +1/2 = 3/4.

Now consider the adapted process v given by V(t) = g(t){t S 1: A B} From I < 1 a.s., it
1
holds that [va (OHZdt < 00 a.s. We have v(t,0)) e Ker[o(t,0))] for all (t,0)). Furthermore,
0
ZV(-) is the martingale zg(r A 13 A .). So v e D.

Finish the example by showing E(Z(1)Z"(1)) = 1. We use the martingale property of

the process Zg(B A .) in the second equality below, and refer to (5.10) in the fourth.

(5.11) E(Z(1)2"<1))= E(Z‘tt A 1)) 2% A 1))) = E(Zf(t A 1)) Zg(B))

=2 jzf(tA13)dP = 1.
{15d}

74

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75

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