WIWINIMIHHWNIHIHllHllWW\WHI

     
   

(9—)
N-P-
4:.

ITH I

 

UBRARY
IMICI‘kuyall State
University

 

 

 

This is to certify that the
dissertation entitled

PARTICLE TRACKING USING STOCHASTIC ,
DIFFERENTIAL EQUATIONS DRIVEN BY PURE JUMP LEW
PROCESSES

presented by

PARAMITA CHAKRABORTY

has been accepted towards fulfillment
of the requirements for the

DOCTORAL degree in STATISTICS

 

 

! .

K S‘s-w! ‘ “up I fi‘}: ' k“ k
3.1AL/(ngqt‘LL. (:21:
Maprl'dféssor's Signature

©Q/3c>/ofi
/ / ‘

Date

~——..

MSU is an Affinnative Action/Equal Opportunity Employer

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5/08 K:/ProilAcc&Pres/CIRCIDateDue.indd

 

 

farm. 3 -

PARTICLE TRACKING USING STOCHASTIC DIFFERENTIAL EQUATION
DRIVEN BY PURE JUMP LEVY PROCESSES

By

Paramita C hakrab orty

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

2009

ABSTRACT

PARTICLE TRACKING USING STOCHASTIC DIFFERENTIAL
EQUATION DRIVEN BY PURE JUMP LEVY PROCESSES

By
Paramita Chakraborty

Stochastic diffusion driven by a pure jump Lévy process is an important core
concept for particle tracking methods used in stochastic hydrology and for tempered
anomalous diffusion models used in (Geo) Physics. In this work we discuss the jump
Lévy diffusion in terms of stochastic differential equations (SDES). We examine the

existence and uniqueness of solutions of stochastic differential equations of the form
dYt = (IO/(5)6125 + b(Yt)dXt

where {X t} is a pure jump Lévy process. Further, we rigorously derive the infinites-
imal generator and the backward equation. It can be shown that the infinitesimal
generator is a pseudo differential operator. Using this form with the backward equa-
tion, we derive the forward equation by an involution type technique. The forward
equation associated with the transition density of the solution process is analogous to
the governing advection-dispersion equation used in particle tracking of heavy tailed

flows and tempered anomalous diffusion models.

DEDICATION

To: Shova Chakraborty, my Mother.

iii

ACKNOWLEDGMENT

I wish to express my deepest gratitude to my advisor Dr. Mandrekar. His
guidance, patience and thoughtfulness made this work possible. It was a privilege to
work with him.

I also thank Dr. Meerschaert for his invaluable advice and encouragement that
enriched my work in many ways.

I would like to thank my thesis committee members Drs. Ramamoorthi, Salehi
and Xiao for their time and interest. I appreciate the help and support of the
Department of Statistics and Probability in last five years.

Finally, I thank my husband, Anindya for being there for me during this journey

and my parents and sister for their unconditional support.

iv

TABLE OF CONTENTS

2.1 Lévy Processes and Lévy Ito Decomposition ........

2.2 Stochastic Calculus w.r.t Le’vy Processes ..........

2.3 Stochastic Diffusion Driven by a Jump Lévy Process &

Existence and Uniqueness of the Solution ..........

3 Derivation of The Infinitesimal Generator, Backward and
Forward Equations ........................

3.1 Properties of the Solution Process ..............

3.1.1 Time Homogeneity of the Solution Process .....
3.1.2 Markov Property of the Solution Process ......
3.1.3 Feller Property of the Solution Process .......

3.2 Infinitesimal Generator and Backward Equation ......

3.3 Pseudo Differential Operator Form of the Infinitesimal

Generator and the Forward Equation ............

4 Application to a Special Case : Diffusion Driven by

an a-stable Lévy Process ....................

4.1 Existence and Uniqueness ...................

4.2 Infinitesimal Generator, Backward and Forward Equations . .

Bibliography ...............................

11

17

20

21

21
22
23

24

31

 

Notations

l.

10.

11.

12.

13.

14.

R : Set of real numbers.

. 1R0: {:1:::I:ER;:B#O}.
. N :2 {1,2,3,...}
. N0: ={0,l,2,3,...}

.R+:={$ER:$ZO}

. CgGR) : All functions f defined on R with compact support and bounded

second order derivatives.

Cm(lR) : m-times continuously differentiable functions on R.

.CO°(R):: 0 0mm)

mEN

. COOUR) :: class of continuous functions on IR vanishing at 00.

(Q, .77, P): Probability space.
L0(Q) : Set of all real valued random variables defined on (2.

50(0, 5, p) : Stable distribution with index of stability at, the skewness param-
eter ,8, the scale parameter a and the shift parameter p.

ForO<a§2,aZO,—1gfiglandpreal.

N (ds,du): Poisson random measure defined on B([O, oo) XR) with mean function

11(ds, du).

q(ds,du):Compensated Poisson random measure defined as :

q(ds,du) : N(ds,du) -— z/(ds, du).

vi

Chapter 1

Introduction

Stochastic diffusion is a very useful tool in hydrology. It is widely used to describe
water flow through a porous medium. The main hypothesis is that the path of a water
(quasi) molecule is governed by a Markov process. This idea has been discussed by
Bhattacharya, Gupta and Sposito [6, 7, 15]. They argued that the trajectory of a
quasi molecule is a Markov process with continuous sample path. With the drift
coefficient chosen to be the drift speed of a quasi molecule and the diffusion matrix
providing a measure for the random variation of the increment of the sample path, the
Markov process can be shown to be the unique solution of Ito’s stochastic differential
equation (SDE) driven by a Brownian motion. Further, the conditional probability
density function of the Markov process solves a forward equation that is analogous
to the advection dispersion equation (ADE) that governs the flow. This mechanical
advection dispersion equation is derived from the Buckingham-Darcy equation [4]
that forms the basis for most current hydrological models of contaminant transport

1

[4, 14].

Recently, it has been observed that this advection dispersion equation may not be
adequate for modeling the heavy tailed contaminant transport in saturated porous
media. For such flows an alternative model has been advocated that describes the dis-
persive flux by a fractional space derivative (the space-fractional advection-dispersion
equation or fADE [5]). A stochastic differential equation driven by a stable-Levy pro-
cess can be associated with this fADE model. If there exists a Markov process that
solves an SDE driven by a stable-Levy noise then it can be shown that the forward
equation of this Markov process is analogous to the governing equation in the fADE
model. This theory has been used by hydrologists as the basis of random walk par-
ticle tracking methods used to solve the fADE to model heavy tailed ground water
contamination transport (Zhang et al [25[,[26[,[27]). The primary purpose of this
work is to lay out a rigorous mathematical foundation for this theory that has been

successfully used in hydrology.

Another example of an SDE driven by a jump Lévy process is the tempered
anomalous diffusion model [10, 20], which has useful applications in (geo)-Physics.
Stochastic diffusion driven by a general jump Lévy process can be used to describe
all these cases. With this motivation we start by deriving the governing forward
equation of an SDE driven by a general jump Le’vy process. Next, as a special case,
we derive the governing forward equation of an SDE driven by a stable-Levy process
that can be used for particle tracking of heavy tailed flows.

2

 

In Chapter 2 we give preliminary definitions and results required for the main
part of the thesis. In Chapter 3 we discuss an SDE driven by a general jump Lévy
process and derive the infinitesimal generator and forward equation. Using concepts
developed in Chapter 3, we derive the governing forward equation of an SDE driven
by a stable-Levy process in Chapter 4.

More detailed discussion and application of the theory presented here can be

found in [11] and [12].

Chapter 2

Preliminaries

The purpose of this chapter is to provide background material for the subsequent

chapters.

2.1 Lévy Processes and Lévy Ito Decomposition

We shall be studying a stochastic differential equation driven by a Levy Process.
This involves defining stochastic integration (in the Ito sense) with respect to a Lévy
process. We begin by defining the Lévy process . Let {9,17, P} be the underlying

probability space.

Definition 2.1.1. A stochastic process {Xt : t 2 0} is a Lévy process if the

following conditions are satisfied:

1. For any choice of n21 and 0 g to < t1 < < tn, random variables XtO’

th‘Xt02 th—th , ., th—th_ are independent (independent increment

1

4

property).
2. X0 = 0 as.

5’. The distribution of X s+t — X 3 does not depend on s (stationary increment

property).

4. It is stochastically continuous, i.e P(|Xt — XSI > c) —> 0 as s —> t for every

e>0.

5. We assume that {Xt,t E R+} is cadalag without loss of generality. (Sato
[23]).
( There is (20 E (F with P[90] = 1 such that, for every w E 00, Xt(w) is

right-continuous in t Z O and has left limit in t > 0.)
Remark:
0) { Xt} is called a Levy process in law, if it satisfies (1), (2), (3) and (4).
(ii) A stochastic process satisfying (1), (2), (4) and (5) is called an additive process.
(iii) An additive process in law is a stochastic process satisfying (1),(2) and (4).

Definition 2.1.2. A probability measure ,u on IR is infinitely divisible if, for any

POSiti've integer n, there is a probability measure #72 on R such that p. : pg.

31%

a) Here pg gives the n-fold convolution of an, i.e, #ii : El, * 'u, * . . . * p.

n

b) We can write pn = til/n and it is uniquely determined. Using this we can define

urn for any rational number rn.

c) For any non-integer t E (0, 00) we can choose a sequence of rational numbers {rn}
such that rn ——> t and define the t-fold convolution of p as pt : lim urn. For

n—+oo

detailed construction and proof of existence of such limit see Sato ([23], page

35).

Observe that by (1),(2) and (3) in definition 2.1.1 above and the fact that for each
t>0,X 2X —X _ + +X2 —Xt +Xt —X forevery n,wecan say
If t [nn1)t it 5 r7, 0
for each t, X t 2 sum of independent identically distributed random variables. Let
,aXt be the measure associated with {Xt} and [An be the measure associated with
{XL — X0}, then “Xi : #n a: #n - - - * pn 2 pg. Thus I‘Xt is infinitely divisible for
7?.

each t. The next theorem precisely gives the relation between a Lévy process and an

infinitely divisible process.
Theorem 2.1.1. ( Theorem 7.10: [23], page 35)

(i) If {Xt : t 2 O} is a Le’vy process in law on IR, then, for anyt 2 0, PXt is

infinitely divisible and, letting PX1 = ,u, we have PXt 2 at.

(ii) Conversely, if p is an infinitely divisible distribution on IR, then there is a Le’vy

process in law {Xt : t Z 0} such that PX1 : p.

An infinitely divisible process can be specified by its characteristic function, which
is given by the Lévy-Khintchine representation as follows:

6

Theorem 2.1.2. (Theorem 8.1: [23], page 37) Lévy-Khintchine representation

(i) If p is an infinitely divisible distribution on le, then its characteristic function

has following representation:

[42) = / emmdw)

: exp [—%(z,Az) + i(’7,z) + [(ei<z’x> — 1 —i(z,:v)lD(:r))1/(d;1:)] (2.1)
le

for z 6 Rd, where D : {:r: : [SCI 5 1}, A is a symmetric nonnegative definite

d x d matrix, 1/ is a measure on le satisfying,
u({0}) : 0 and [flat]2 /\1)1/(d:r) < 00
Rd
and 7 E Rd. The representation of [t in (i) by A, V, and 7 is unique.

(ii) Conversely, if A is a symmetric nonnegative-definite d x d matrix, I! is a measure

as above, and ”y 6 Rd, then there exists an infinitely divisible distribution ,u

whose characteristic function is given by (2.1).
Remark: We call (A, 11,7) in the above theorem the generating triplet of p.

In this work we express the stochastic differentiation w.r.t a jump Lévy process
as the stochastic differentiation w.r.t a compensated Poisson random measure. To
understand this representation we need to know the decomposition of general Lévy

7

processes in terms of integration w.r.t Poisson random measures. This is called the
Levy-Ito decomposition.

We start with the definition of Poisson random measures. Let

2+ 2 {0,1,2,3,..}U{+OO}

Definition 2.1.3. Let (G, B, p) be a o-finite measure space. A family of Z+-tralued
random variables {N (B) : B E B} is called a Poisson random measure on G with

intensity measure or mean function p, if the following hold:

(1) for every B, N(B) has Poisson distribution with mean p(B);
(2) if Bl, ..., Bn are disjoint, then N(Bl), ..., N(Bn) are independent,-

(3) for every (.2, N(.,w) is a measure on 6-).

The random measure q defined by q(B) = N (B) — p(B) is called the compensated

POiSSOTl TUTLdO’ITl measure.

The next theorem gives a constructive decomposition of a Lévy process as a
sum of a jump part and a continuous part. In this theorem we shall use stochastic
integration w.r.t a Poisson random measure (see Section 2.2 for precise definitions).

Let H = (0,00) x (le \ {0}). The Borel o-algebra of H is denoted by B(H).The

basic decomposition theorem is given by,

Theorem 2.1.3. ( Theorem 19.2 : [23] page 120)
Let {Xt : t 2 0} be an additive process on le defined on a probability space (9, f, P)

8

with system of generating triplets {(At,z/t,ryt)} and define the measure 17 on H by
17((0, t] x D) :- I/t(D) for D E B(le). Using 90 from definition(1.1) of an additive
process, define for B E B(H),

#{s: (s,Xs(w) — X3_(w)) E B} ,for w E 90,

0 ,for wfiéQO,

J(B,w) 2

Then the following holds:

(i) {J (B) : B E B(H)} is a Poisson random measure on H with intensity measure

~

V.

(ii) There is 91 E f with P[91] = 1 such that, for any or E 91,

X§1)(w)_—_1€ir3 / {st(d(s,a:),w)—xz7(d(s,:v))} + / xJ(d(s,:1:),w)
(0,tl><{€<|1'|<1} (Ottlelxlzll

is defined for allt E [0, 00) and the convergence is uniform in t on any bounded

interval. The process {Xt(l)} is an additive process on le with {(0, Vt,0)} as

the system of generating triplet.

(iii) Define, Xt(2)(w) = Xt(w) — X§I)(w) for w E 91.
There is 92 E f with P[92] = 1 such that, for any a) E (22, Xt(2)(w) is contin-
uous in t. The process {X P} is an additive process on le with {(At,0,’yt)}

as the system of generating triplet.

(iv) The two process {Xt(1)} and {Xt(2)} are independent. {Xt(1)} is called the jump
part and {X§2)} is called the continuous part of the process {Xi}.

9

Finally the Levy-ltd decomposition of a Lévy process is given as follows:

Theorem 2.1.4. (Theorem 2.4.16:[23],page 108) (The Le’vy-Ité decomposition)

If X is a Levy process in le with generating triplet (A,1/,7), then there exists a
Brownian motion BA with covariance matrix A and an independent Poisson random
measure N on IR x (le \ {0}) with intensity measure 17((0,t] X B) = t x 11(8) for

B 6 EURO), such that, for each t Z 0,

X(t) : 7t + BA(t) + / xq(t,dx) + / xN(t,dx)
[x[<1 |x|21
where q is the compensated Poisson random measure associated with N as in Defini-
tion 2.1.3. The Brownian motion part gives the continuous part of the Lévy process
whereas the integration w.r.t Poisson and compensated Poisson random measure con-
tribute to the jump part. In the next section we shall formally define integration with

respect to a Poisson random measure and a compensated Poisson random measure.

In this work we shall consider a pure jump Lévy process with A = 0 and 7 = 0,

i.e a Lévy processes of the form:

X(t) = / xq(t,dx) + / xN(t,dx)

[x[<1 [11:]21

10

2.2 Stochastic Calculus w.r.t Lévy Processes

In the previous section the Lévy-Ité decomposition shows a Lévy processes can be
expressed in terms of a Brownian motion and a Poisson integral. However, here
we shall be considering only pure jump Lévy processes. Therefore for the main
results we shall ignore the Brownian motion part and concentrate solely on stochastic
integration w.r.t a Poisson (compensated) random measure. We shall follow the
concepts discussed by A.V Skorokhod in [24] to define stochastic integration w.r.t a
compensated Poisson random measure.

Let us consider 0 S to < T < 00 and let V be a measure defined on [t0,T] x R.
Let B be the ring of all Borel sets A in [t0,T] x IR for which 1/(A) < 00.
Let N be a Poisson random measure defined on B with intensity measure V. For

every A E B we shall denote by q(A) the random measure defined by the relation:

We suppose that for every t 6 [to, T], a o-algebra ft of events A is defined, such that
A C [t0,T] x R (that is, (s,u) E A only ifs E [t0,t]). Also we assume that N(A)
is measurable w.r.t ft. Also, for all sets A1,A2, . -- ,Ak in [t,T] x IR the quantities
N (A2) are mutually independent of any event in ft.

We shall use the following function spaces in our definition:

M(ft) = {f(t,u) : f(t,u) is a random function measurable w.r.t ft}.

11

We shall call the function f(t,u) a step function if there exists to < t1 < <

tn 2 T and Borel sets B1, Bg, - -- , Bn in IR such that f(t,u) is constant on every set
ltkatk+1l x Bj, k = 0,1 (n— 1), j = 0,1 n, and UBj : IR. We set the

following:

M003) 2 {step functions f(t,u) in MUQ) : 3 e > 0,f(t,u) = 0 for [u] S e}.

T
A7151) 2 E[f(,)(tu|z/ dt ,du) <00}.
w
T
MI?) : f(,tu) ):P[ /|f(,)(,tu|udtdu)<00[=1}.
to R
T
M51) : u):/ E|f(,)tu|2u (dt,du) <00}.

H-

qa\,

0

Mi” = {f(t,u) P

fl.‘ I

u)2z/ u =.
0M )|(dt,d)<00[ 1}

Definition 2.2.1. Stochastic Integral w.r.t compensated Poisson random measure

(See (24/, page 35-37):

(a) Assume f(t,u) E MOU't). Taking uj E Bj, we define

T
f/f(t,u>q<dt,du) = Z f(tkauj)9(ltkatk+1lXBj)-

to R 0.<_ks(n—1)
ISan

12

Remark : Note that in this case it can be shown that the following [to isometry

holds: 9
T “ T
Ef//f(t,u q(dt, du)=//E[f(t,u)|21/(dt,du)
toR t0 R

(b) It can be shown that For every f(t, u) E Nfé1)(ft), there exists a sequence of

functions fn(t,u) E N10(ft) fl Mglnft) such that

T
lim //E[fn(t,u)——f(t,u)|21/(dt,du) = 0

n—+00
to R
T
Hence the sequence of random variables I f fn(t,u)q(dt,du) will converge in
tolR

probability to some particular random variable which we shall denote by:

T
tféf(t,u)q(dt,du), whenever f(t,u) E MgDU-ft).
0

(c) Now let f(t, u) E M? )(ft), define gn(x): 1 for [x] S n and gn(x): 0for

[x] > n. Then for all n E N,

T
fn(t,U) = f(t,U)9n('//|f(t,U)|2q(dt,dU))

0 IR

— (1) . T .

belongs to Mq (ft), and consequently the expression f f fn(t,u)q(dt,du) is
tolR

13

meaningful from (b). Also it can be shown for n’ > n,

P [f/fn;(t,u)q(dt,du) — [T/fn(t,u)q(dt,du)| >0]

to R to IR
i/
g P [f(t,u)[2u(dt,du) >72]
[I0 R

Since the probability on righthand side approaches to zero as n —> 00, the

T
sequence of random variables f f fn (t, u)q(dt, du) will converge in probability t0
to IR

 

T
some particular random variable which we shall denote by f f f(t, u)q(dt,du),

t0R
whenever f(t,u) E Mé2)(}'t).

Next, we give the definition of a stochastic integral with respect to a Poisson

random measure N.

Definition 2.2.2. Stochastic Integral w.r.t Poisson random measure (See (24/, page
38-41).
592(7) T

t),henfff(,)qtu d(t, du))andff(t,u)1/(dt,du)
tolR

both are finite. We set for f(t,u) E Mé2)(ft) fl thft)

f/faw (,=dtdu) f/f(,)qtu (dtdu)+/f(,)tuz/ (,)dtdu

tOIR t0R

(a) Iff(t u) e M‘2 (a) 0M

(b) For every function f (t,u) E MI?) (E), it is possible to construct a sequence

14

fn(t,u) E M£1)(.7:t) fl Mzgghft) such that

nlim OO/T/EIfM t ,u)— t u)[1/(dt du)—— - 0

to IR
and therefore
T
"Ii%//E[fn(t,u) — fl(t,u)[1/(dt,du) = 0
1600“) IR
T
Hence the sequence of variables I f fn(t,u)N(dt, du) will converge in probabil-
toR

ity to some limit, which we shall denote by

T
f/f(t,u)N(dt,du)

tolR

(c) For f(t,u) E Mflm). Then for every n E N,

T
fn(tIU) = f(t,UIgn(/|f(t,UII2V(dt,dU))
0

will belong to M181)(.Ft), and for n’ > n,

T T
Ptof/fn/(tmwdtdu)—t[1{fn(t,u)N(dt,du) >0

15

g PLZR/lf(t,u)|zx(dt,du) >n .

T
Consequently ”limoo f f(t, u)N(dt, du) exists in the sense of convergence ofprob-
t

ability. We shall denote this limit by,

T
[f(t,u)N(dt,du) for f(t,u)eM§Q)(J-'t).

t0
Now we can proceed to define stochastic integration w.r.t jump Lévy processes.

Definition 2.2.3. (Stochastic Integral w.r.t Lévy processes).

Let {Xt} be a pure jump Le’vy process taking values on IR with generating triplet
(0, V, 7). We want to define stochastic integration of the form Yt = Y0+f6 L(s)dXs.
The Lévy-Ité decomposition shows that {X t} can be expressed in terms of a Poisson

integral. Using that form we can define:

where we assume to = 0 and L(t) E Mé2)(ft)[u:0.

In general a stochastic process taking values in IR is called a jump Levy-type

16

st ochastic integral if it can be written in the following form:

t t t
=I/(3(.9+/G )ds+/ / H( (s ,(x)q ds, dx) )+/ / K(8,$)N(d3ad$)
0 0 0

l$l<1 |I|.>_1

where [G'Il/2 E Mé2)(ft)[u:0 ; H(t,-) E My)(ft) and K predictable. Here N is
a Poisson random measure on (IR+ x R0) and q is the corresponding compensated

Poisson random measure.

2.3 Stochastic Diffusion Driven by a Jump Lévy
Process & Existence and Uniqueness of the

Solution

Let {Xt} be a pure jump Lévy process with generating triplet (0,1/,0) of following
form

X(t): / xq(t,dx)+ / xN(t,dx) (2.2)

0<|x|<1 |x|21
where N is a Poisson random measure with intensity measure 17((0, t] x D) = tl/(D)

for D E 3(R0).

Let us consider the drift coefficient functions a and the dispersion coefficient
function b, where a, b : IR —> IR. We are interested in the diffusion equation driven

l7

by pure jump Lévy process {Xt}, i.e, SDE of the form:
dYt = a(Yt)dt + b(Yt)dXt (2.3)
Using the Lévy-Ité decomposition of Xt we can rewrite (2.3) as :
dYt = a(Yt)dt + / b(Yt)xq(dt, dx) + / b(Yt)xN(dt,dx) (2.4)

0<|x|<1 [lel

Thus, we get a special case of the (jump) Levy-type stochastic difierential equation.

Under certain conditions on the coefficient functions a and b of the diffusion
process, we can show that there exists a unique solution process to the diffusion
equation (2.3). The general stochastic differential equation w.r.t a Lévy process with
the existence and uniqueness of the solution is discussed by D. Applebaum ([1] :
section 6.2). Therefore for the following theorem we give just a sketch of the proof

and refer to [1] for details.

Theorem 2.3.1. Let coefi‘icient functions a and b of diffusion equation (2.3) satisfy

the following growth condition and Lipschitz condition

(A) Growth condition: there exists constant C > 0 such that \7’ y in IR

la(v)l2 + Ib(y)|2 .<_ C(1+ In?)

18

(B) Lipschitz condition: there exists constant C’ > 0 such that V y1 y2 in IR
9
MM) - avg»? + lb(yi) — but? 3 d(t/1 — yzl“)

Then, there exists a unique solution process Yt for the equation (2.3); also the solution

process is continuous w.r.t the initial value.

Proof. To prove the theorem use the alternative form of the diffusion process as in
(2.4). The proof is done in two steps. In the first step consider equation (2.4) up to

small jumps only, i.e consider

dZt : a(Zt)dt + / b(Zt)xq(dt, dx)
0<|x|<1

Such a process {Zt} can be constructed using Picard’s method (Skorokhod [24]) and
using conditions (A), (B). Further it can be shown the solution {Zt} is unique.
In the second step the large jumps can be added to {Zt} using an interlacing tech-
nique (Ikeda and Watanabe [17]) and hence the solution process {Yt} for the main
equation (2.4) is constructed. Uniqueness follows from uniqueness of {Zt} and the
interlacing structure.

For detailed constructions and proof of uniqueness see [1] (Theorem 6.2.3,
page 304 & Theorem 6.2.9, page 311). For the proof of continuity w.r.t initial value

and Markov property see [24]. CI

19

Chapter 3

Derivation of the Infinitesimal
Generator, Backward and Forward

Equations

In the previous chapter we observed that, under certain restrictions on the drift and
the diffusion coefficients, a unique solution process {Yt} of the diffusion equation in
(2.3) exists. Further, the solution is a Markov process. In this chapter we derive
the infinitesimal generator and associated backward equation of the Markov process
{Yt}. Using Fourier analysis concepts we can show that the infinitesimal generator is
a pseudo-differential operator. Then we derive the governing forward equation from

the backward equation using an involution type technique.

For simplification of calculations we restrict the study to one dimension and also

20

 

I‘

to the case where drift and diffusion coefficients are only space dependent. The

theory can be easily extended to an SDE with time homogeneous coefficients defined

on W.

3.1 PrOperties of the Solution Process

3.1.1 Time Homogeneity of the Solution Process

Let us consider the SDE in (2.3) with a little modification. Consider
dYt = a(Yt)dt + b(Yt)dXt (3.1)

for t 2 3, given Y3 = y and X 3 is a jump Lévy process.
Also, we assume coefficient functions a and b satisfy growth condition and Lips-

chitz condition from Theorem 2.3.1.

Denote the unique solution of (3.1) by Yt = Yty,s . Then,

t+h t+h
33% : y + f a (Yé’f) du + f b (Yff’t) qu
t t

= y + foha (32%) d” + [Ohb (YEW) CD?”

with u = t + v and Xv = Xt+v — Xt. On the other hand,

Y5’0 = y + tha(Y5"0) dv + fotb(Y,§”O) dXv

21

Now from the definition of Lévy processes, {Xv} and {X 1,} have same X O-distribution.
. . . . ,t
Then it follows from the uniqueness of the solution for SDE in (3.1) that {YtgihhtZO

and {YE’O}, h 2 0 have same I’D-distribution; i.e, {YtItZO is time homogeneous.

3.1.2 Markov Pr0perty of the Solution Process

Let {7}} be the o-field generated by {X3 : s S t}. Let {Yt} be the solution process

of equation (2.3). From the equation we know that Yt is ft measurable.

Theorem 3.1.1. If there exists a unique solution of SDE (2.3), then the solution

process is a Markov process.

Proof. By construction we can re—write Yt+s as follows:

Y7+3==Y}—Ffi40$t—fs)

where
t+s t+s
M(s,t + S) : / a(Yv)d’U + / b(Yv)dXv
s 3

Following the Lévy-Ité decomposition M(s,t + s) is a{N(v, A) - N(u, A),s S u <
v S t,A E B(IR0)} measurable ( EURO) is the Borel sigma field of R0). Clearly
M(s,t + s) is independent of .7-"3 Q o{N(u,A),u g s,A E B(IR0} and that leads to

the Markov property. See [24] (page 75, Theorem 1) for a more detailed proof. CI

22

3.1.3 Feller Property of the Solution Process

Let {Yt} be the solution process of equation (2.3). Let {Yty} be the solution process
with initial condition y. We want to show that the solution process is continuous in
terms of initial value. Let functions a and b be the drift and diffusion coefficients as
in (2.3). We assume the coefficients satisfy the growth condition in Theorem 2.3.1.

Define
t

t
2,9 =y+ [a(zg’)ds+ / / b(Z3y)uq(ds,du)
0 0<[u[<1

where compensated random variable q is defined as in (2.4). We want to show that

2
if yn —-> y, then Z?" 9+ Z? for 0 < t < T. Now

2
t
K)
IZty" — Ztyl“ S 3|yn - :yl2 + 3 f[a(Z§’”) — a(ZSy)]ds
0

t 2
+ 3 f / [b(Z§’") — b(z§’)] uq(ds, du)
0 0<[u[<1
If we write f(,, = f u21/(du), then using Cauchy-Schwarz inequality and Doob’s

0<[u[<1

martingale inequality in the right hand expression above, we get:

t
E( sup [an — 2,9?) g 3|yn —- y|2 +3T/E( sup [a(zg") — a(zgl)|2)ds
0<t<T 0 0<t<T

t
+12RV/E( sup [b(Zgn)_b(Zg)l2)ds
0<t<T

23

By the growth condition

t
E( sup [an — 2,342) g 3|yn — yl2 + CT/E( sup E [23” — Z§|2)ds
0<t<T 0 0<t<T

where CT is a constant that depends on T and constant C, in growth condition

given in theorem 2.3.1 only.

Hence by Gronwall’s lemma we can say, if yn —> y then E( sup [ngn — ZtyIQ) —> 0.
0<t<T

Adding the large jump part to {Zt} by interlacing we can get {Yt} and thus the

process {Yt} is continuous in terms of initial value.

3.2 Infinitesimal Generator and Backward
Equation

Let P(t, x, y) be the transition probability for the Markov process Yt. That is, for any
A E B(IR), (B(IR) is Borel sigma field oflRO), we have P(t,x, A) = Pg;{Yt E A}. Here,
Pg;(-) should be interpreted as conditional probability with given condition Y0 = x.
The transition probabilities of a Markov process defines a transition semigroup; and
the semigroup uniquely defines the infinitesimal generator. Let us start with the

definition of a semigroup.

Definition 3.2.1. A family {Tt;t 2 0} of bounded linear operators on a Banach
space B is called a strongly continuous semigroup if

24

(a) TtTS : Tt-I—S f0?" ta 3 E [03 00)

(c) ltilrthf = ffor anyf E B.

Definition 3.2.2. The infinitesimal generator of a semigroup {Tt} is defined by the
formula
Thf - f _

A :l'm
f rho

Its domain D A consist of all f E B for which the above limit exists.

Let C be the space of bounded continuous functions on IR. For f E C define the

following operator:

are) = [ f(y)P(t,x.dy) = Eflf(Yt)l

This infinitesimal generator is also called the infinitesimal generator of the Markov
process {Yt}.

In section 3.1 we already observed that the solution process {Yt} of the SDE in
(3.1) is a Markov process. In this section we shall derive the infinitesimal generator
of the solution process. We shall use the Ité formula for stochastic integrals with
respect to Poisson random measure. The Ité formula for general Lévy type integrals
is given by Ikeda and Watanabe [17]. Using their theorem we can state the Ité

formula for our case as follows:

Theorem 3.2.1. ([17] Theorem 4.1, page 66): Let {Yt} is a pure jump Lévy type

25

integral, i. e,

Y(t).=Y(0)+/tG(s)ds+/Ot / H(s,x)q(ds,dx)+/Ot / K(s,x)N(ds,dx)
O

0<[x[<1 [xIZI

where N is a random Poisson process on ([0, 00) x 1R0) with intensity measure 17, with
H (t, -) E Mé2)(.7:t) and K predictable. Let C2(IR) be the set of twice diflerentiable

functions on IR. Then for any function F E C2 (IR) following holds:

Let us assume the coefficient functions a and b satisfy growth condition and
Lipschitz condition as in Theorem 2.3.1. Then a unique solution process {Yt} for the
SDE in (3.1) exists and it is a Markov process. For f E C if we define (T t f)(x) :—

E” f (Yt), then the infinitesimal generator of Tt exists.

Note that equation (3.1) or (2.3) has alternative form as a pure jump Lévy type

26

SDE as in equation (2.4), so we can express Yt as :

t t t
Yt— — Y0+/a(YS) ds+/ /( b ()YS )xq( (ds d,x)+/ b (Y3) )xN( ds ,dx) (3.2)
0 O O

lr|<1 le21

where N is Poisson random measure with intensity measure 17(dt,dx) = dtI/(dx).

Equation (3.2) with Theorem 3.2.1 gives us the infinitesimal generator as follows:

Theorem 3.2.2. (Infinitesimal Generator): Let {Yt} be the solution process of
the stochastic difierential equation: dYt : a(Yt)dt + b(Yt)dXt, where {Xt} is a real
valued pure jump Le’vy process with generating triplet (0, 11,0). Also, suppose that the
coefi‘icient functions a and b satisfy the growth condition (A) and Lipschitz condition
(B) as in Theorem 2.3.1.

If Cu 2 flib‘IZl xu(dx) < 00, then for any function f E CgflR) the infinitesimal
generator A of {Yt} is given by:
AM) = r’(y)(a<y) + claw) + ] {f(y + but.) — f(y) — f’(y)b(y):v]v(d:v) (3.3a)

IR0

alternatively, if Ky : f0<|x|<1$V(dx) < 00, then for any function f E C8(IR) the
infinitesimal generator A of {Yt} is given by:
AM) = f'(y) (a(y) — Kym») + ] {f(y + bum) — f(u)]l/(d-r) (3%)
1R0

27

Proof. Let us assume Y0 = y. Using Theorem 3.2.1 for any f E CgUR),

f()/I) - f(y) t
: ff,(Y3)a(Ys)d3 +/ {f(Y(s—) + b(YS):L‘) - f(Y(S—)) }N(d3, (1:17)
0 0 le21

 

Af(y) = 13er t
1 t
_ - _ y I
— ltlflI tE ff (Y3)a(Ys)ds

 

28

 

 

Now let us write

Cu 2 / xu(dx).
Iévlzl

 

Then
I

my) (a(y) — Kym) + fR0{f(y + bun) — r<y)}u(dx>,
Af(y) = <

my) (a(y) + Cum) + fR0{f(y + beam) — f(y) — f’(y)b(y):v}1/(drv),

if CV < 00.
This completes the proof. Cl
Remark:

i) In case CV and K V both are finite we can use either form because both forms will

be equivalent.

29

ii) In case neither of CV and Ky are finite we can use equation (3.4) form of the

infinitesimal generator.

We can show, for the infinitesimal generator of any Markov process, in particular
for the solution process of the SDE driven by pure jump Lévy process the following

backward equation holds.

Theorem 3.2.3. (The backward equation) : Let A be the infinitesimal generator
as in Theorem 3.2.2 for the solution process {Yt} of the SDE (3.1). Let f E C3(IR).

Define, u(y,t) : Ey [f(Yt)]. Then, %% exists and

Bu
BT = A(u) (3.5)

where the R.H.S is to be interpreted as A applied to the function y H u(y,t).

Proof. Let g(x) : u(x, t). Then, using Markov Property,

Eyller2I—9W) : %{E9[EY"(9(Yt))] — Eyl9(Ytll}

 

= §{Ey[Ey(gm+r>Ifr>i - E” W01}

= ;,1‘Eyl9(Yt+r) - g(YtII

H(y,t+r) — H(yi) _) 9383,10.

r 0t

 

Eyl9(Y-r)I - 90/)

7.

Therefore, A(u) = lim,.10 exists, and 53%” = A(u). Hence the

backward equation. CI

30

3.3 Pseudo Differential Operator Form of the
Infinitesimal Generator and the Forward
Equation

Using Fourier analysis we can show that the infinitesimal generator in (3.3) is a
pseudo-differential operator (in sense of Jacob [18], definition 3.3.3) defined on the
anistropic Sobolev space H€2t2(IR). Here we show that the transition probability
density function of the solution process satisfies a deterministic differential equation
viz. the forward equation. The forward equation can be derived from the backward
equation using the infinitesimal generator in its pseudo differential operator form.
This forward equation gives the governing equation of diffusive flows and thus vali-
dates the key role of jump Lévy SDE in stochastic modeling of anomalous diffusion.
To derive the forward equation we assume the density function of the solution process
belongs to the anistropic Sobolev space H €2,2(IR). For this section we are going to

use the Fourier transform of a real function f as follows:

00

f(€)=F(f(€))=(2r)"1/2 [ immune

—00

We shall discuss other required concepts and definitions as we proceed.
Lemma 3.3.1. Let {Yt} be the solution process of the SDE in (3.1). Let A be the

infinitesimal generator for {Yt}, given as in Theorem 3.2.2. Let us make a change

31

of variable b(y)x 2 —v in (3.3). Let J(v) be the Jacobian of the transformation. We

define V1(y, dv) 2 “MI/(€65). Then A can be expressed as follows:

Af(y) = B(y)f'(y) + [ [f(y - v) - f(y) + f'(y)Tv|U|§]I/1(y.dv) (3-6)
R0

,2
[a(y) + Cyb(y) + fRO u(l—l_:[—v|2)l/1(y,dv)], if CV < 00;

a(v) - Kub(v) - IRO(W)V1(y,dv)], z'f Ky < 00.

where B(y) 2

Proof. From Theorem 3.2.2, if we consider form (3.3a), the infinitesimal generator

can be written as:

Af(y) = my) (a(y) + aux) + [ {f(y + b(y)x) — f(y) — r’(y)b<y>x}u(dx)
IR0
Let us consider the change of variable b(y)x 2 —v. Letting J(v) be the Jacobian of
this transformation we define : V1(y, dv) 2 J(v)1/ (fig—1;). Then

Ara) = f’(y)(a(y)+Cub(y))+ [{fu—v)—r(y)+f’(y>v}u1(y.dv)

= f’(y)[a(y)+Cub(3/)+ ] v(

R0

+ {f(y —' )— M) + f’(y)fi}vi(y,dv)

)Vl (31,010)]

1+ [v]2

v
1+ |v|2

 

] 1/1 (y, aIv)

2
Where Bu) = [a(y) + Cuba) +Rf (fill—[12)V1(y,dv)]-
0

Ram Theorem 3.2.2, if we consider form (3.3b), the infinitesimal generator can be

written as

Again change of variable b(y)x 2 —v gives,

Af(y) = f’(y)(a(y) — Kub(y)) + [{f(y — v) — f(y)}u1(y, du)

 

= Buu’u) + [ [f(v — v) — f(y) + my) 1 +"lv|2]u1(y,dv>
1R0

Where B(y) = a(y) — Kub(y) — f —L,7 Vite/adv) -
1R0 1+[il
Hence the lemma is proved.

Proposition 3.3.1. (Le’vy Khinchin representation, see [18])

We say If) : IR ——> (C is a continuous negative definite function, if w has the following

i"€19resentation:

 

w(£) 2 c + dig + q(t) + [(1 — e435 — ixé )z/(dx)

3.7
1 + [x]2 ( )
R0

with c > 0, d E IR, q is symmetric positive semidefinite quadratic form on IR and V

33

is the Lévy measure associated with if: such that,
[at]? A1)z/(dx) < 00
1R0
and 7,!) is uniquely determined by (c, d, g, V).
Definition 3.3.1. We call a function Q : IR x IR —> C a continuous negative definite

symbol if Q is locally bounded and for each x E IR the function Q(x, .) : IR ——+ C is

continuous negative definite.

Definition 3.3.2. We define the Schwartz space S(IR) as all functions u E COO(IR)

Such that for all m1, m2 E No

pmiim2(uli: 22%[UH13I2I71 Z Iaku($)l] < 00

16sz

The pseudo-differential operator associated with the symbol Q(x,§) are defined

as follows:

Definition 3.3.3. For a continuous negative definite symbol Q(x,§), we define the

PSeudo-diflerential operator Q(x, D) by :
Q(x,D)u(-'r) z: (2r)‘1/2[ei$5Q(x.€)fi(€)d€ (3.8)
IR

fOru E S(IR).

34

The next theorem gives the pseudo-differential operator representation of the
infinitesimal generator. Consider an SDE of form (3.1). For the coefficient functions

a and b let us define

M(x) z: max{la(rc)|. tori} (3.9)

Theorem 3.3.1. Let us use the measure V1 and coeflicient function B from Lemma

3- 3.1 to define following continuous negative definite symbol:

 

Q(x,€) 2 R[(1— e—ivé — 1f0i|2)1/l($,dv) — iB(x)§ (3.10)
0

Let A be the infinitesimal generator defined in section 3.2. Then the infinitesimal
generator, restricted in 8 (IR) is a pseudo-differential operator with negative definite

Symbol Q as above. That is :

Ar(x)=—Q<m,D>r(x) ~—— _(2..)-% [armament (3.11)

where f is the Fourier transformation of f E 8 (IR)

Proof. To prove this theorem we need the following bound of symbol (3.10).

Lemma 3.3.2. Let the function M () be defined as (3.9). For the continuous negative

definite symbol Q(x,£) given in (3.10), then for some constant c

|Q(rv,€)l s cM(:c)(1+€2)- (3.12)

35

Proof. Consider Q(x,§) given in (3.10).

[2
First, consider the form B(x): a(x ) + Cyb(x +RvaI—H—IZV1(y,dv)

 

aao=](v+r“— w€)mauv—ka

 

1 + [v]2
R0
_ —Z’U€ ' ’U ' I’Ul2 - D
_/ 1—e —21+lvl2€_w—1_fl7[2€ V1(.’L‘,d‘U)—’t a(x)+CV (x)€
IRo
:: /(1 — e—ivé — ivg) 111(16, dv) — i [a(x) + CVb(x)]£
180
Using reverse transformation b(x)y 2 —v

8W3) y€ + ib(x)y €>V V01?!) — i [0(1) + CVb(x)[€

link
2R:1( —e Wivhaa) )y§)1/(dy)—i§[a(x)+b(x) ] yu(dy)[
=0/1
yl<

lylzl

_Hz-b(x)y£+,-b( )yg)u(dy)+ ] (l—eib(x)y€)u(dy)—i§a(x)

0<l||y|21

Next, consider the form B(x) 2 a(x) — f(,/b(x) -— f Div—[2V1 (y, dv).
IRo

 

ano=](vwd%— w€)aeao—wux

R0 1 + [v]2
z [(1— (”5) V1(x, dv) — i[a(x) -— Kub(:c)]€
R0

36

Using reverse transformation —b(x)y 2 v as the previous case,

Q(:v,€) = ] (1 — eib<x>y€)u(dy> — 2' [am — mm]:

R0

: / 1 _ eibfoy€ + ib(x)y§) V(dy) + f (I _ eib(1’ly£)1/(dy) — i€a(x)
0<|yl<1 |y|21

Thus, for both forms of B () we can write:

62013.5): [ (1—eib($>y€+ib(x>ys)u(dy)+ [ (l—eibixiyi)u(dy)—zra(x>

0<Iyl<1 Iyl21
(3.13)

Further, note that we can get the following bounds :

[1— etblxiyi + than: s lb($)£vl2 and |1 — eibixiyél s 2

 

also, [5] _<_ (1 + [5|2). Recall that V, the Lévy measure for Q(x, D) in (3.10),
satisfies f [yl2V(dy) < 00. Therefore, since f V(dy) < 00 we have the

0<|y|<1 Ilel
following:

IQ(:v,£)l _<_ 15:21me [ lyl2v(dy)+ [ 2V(dy)+|a(:v)ll€|
0<lyl<1 lylzl
s cM(:v)(1+|€|2)

where M(x) 2 max{|b(x)|2,a(x)} and c is a constant that depends only on V. CI

37

Proof of Theorem 3.3.1 continued
We shall use the notation f 2 F ( f) to denote Fourier transform as defined in the

beginning of this section. Also recall that, the inverse Fourier transform of a function

g is given by F—lgm) .—_- (2704/2 cf eixng(x)dx and that F(g("))(k) = (am
—00

where g(n) is the nth-derivative of g, n E N .

Using Lemma 3.3.2 we can say for f E 8(IR)

[[ gramme] s. cM(:v)[(1+€2)|f(€)ld€
M(x) [ |f(z)| + lf”(z)ldz

I/\

Since the functions in 8 (IR) are rapidly decreasing, it is easy to see that f, f ’ ’ E L1 (IR)
(see the norm used in Definition 3.3.2). This justifies the use of the Fubini theorem

the next steps of the proof. By Lemma 3.3.1,

—Q(rv.D)f = —<2w)*i [ eix€Q($,€)f(€)d€
: —(2rr)_2 ] eixé [/ (1 — e-ivé _ 1:”; I2)u1 (x, dv) — racing] f(g)dg

IRo

= - ] [op—i ] ei$€F(r(t>-f<€—v)—1+]vlgr’(o) d€](,u1xdv)

R” view an

]:1(x(.) dv)+ B(a: )r’ (x)

 

 

v
x _—
1+ [v]2

38

Hence, Theorem 3.3.1 is proved. CI

Next,the we shall discuss spaces associated with the pseudo differential operators.
For a real valued continuous negative definite function w(§) and s _>_ 0 let us define

a norm

Halli), == [l1+ gangsta

An anistropic Sobolev space with a negative definite function w is given by
Hitachi) :2 {u e L2(IR) : Maui’s < 00}

These are Hilbert spaces under norm “all?” and arise naturally in the discussion of
the pseudo-differential operators (See [16, 19]). Jacob and Schilling [19] showed that
with appropriate choice of 2]), a pseudo-differential Q(x, D) operator associated with
the generator of a Lévy type processes maps the space Ari/1.8+? to H If”? and hence
using Sobolev’s embedding theorem, Q(x, D) can be extended to COO(IR). Thus we
can say the pseudo differential operator representation of infinitesimal generator A
can be extended to COO(IR).

In our case we shall use the Sobolev space H€2’2. In most cases, the density of a
Lévy type processes belongs to this class, for example use the stable characteristic

2
function to see that the stable-Levy density belongs to H ‘5 ’2.

Theorem 3.3.2. (The forward equation)
Let {Yt} be the solution process for the SDE in (2.3). Let us further assume that

39

there exists a transition probability density for Yt and that px(t,y) is the density of
Yt, given Y0 2 x. We assume p1.(t,) is in H€2’2 for all x E IR.

Let us make a change of variable b(y)x 2 —v in (3.3). Let J(v) be the Jacobian of the
transformation. We define V1(y,dv) 2 J(v)V(b:(%‘)—’). We consider the case when the
measure V1 is of the form V1(1‘,dy) 2 h(x),u.(dy), where h is a measurable function
on IR and [x is a measure on IR.

For coefi‘icientfunctions a and b in the SDE (2.3) define M(x) :2 max{[a(x)|, |b(x)|2}.

Let us assume the coefficient functions are such that
[ M2(x)dx < 00 (3.14)
IR

Then the transition probability density function satisfies the following equation:

’lny <00,

033mm) = [ [rpm - hrs, y - r) - (p. - h)(s, y) + rum - h>’(s, y)]#(d(-7‘))
R0
0

— 55(1):: ' (3(3)?!) (3-153)

where (Pa: - h)(s, v) = h(y)p:v(s,y) and 0(9) == 0(9) + Cub(y).

Alternatively, if K V < 00,

_2

anyx-Hfisay) (3.151))

319mm: ] [(Px-h)(s,y—r)-(px-h)(s,y)]#(d(-r))

Os
R0

40

when? (Px - h)(8, y) = b(y)px(s. y) and H (y) = a(y) - K ub(y)-

Proof. Let p3,.(t, y) be the transition probability density of Yt starting at Y0 2 x. Let

uO E L2(IR) be a twice differentiable bounded function. Let us write

u(x,» = Exluthll = ] ”0(ylpx(tay)dy-
R

Then Au(x, t) 2 6 u(x, t) is defined. Since uo E L2(IR) we can have a constant c’ so

that following holds:

um) = Ema/rm): [uo(y+a:>po(t.y)dy
R

u0(z)p0(t, z — x)dz
2 21(5, t) 2 u0(z)p0(t, z — §)dz
”Wk-”£1500, -93.

=> |fi(€,t)l2 s luo(z)l2|fio(t,—€)|2dz

fi\ fi\ %\ W\

a was)? s c’lp‘o(t,€)l2 (3.16)

If we assume the transition density function vanishes at t 2 00, then integration by

parts gives

00 00 a 00 00 a
f [ —U(v,t)px(t,y)dtdy = - f / n(y,t)—pa:(t,y)dtdy (3-17)

—00

41

Now substituting the backward equation in the left hand side we get,

(9
[:0 [00 P2201304 u(y,t)) dtdy= —[_: [00 u(ty, )apatyIdtdy (3.18)

The operator A acts on u(y,t) as u being a functions of y. Consider the integration
part with respect to y in the left hand side of (3.18) and to simplify the notations

we ignore the other variables in the term for next steps of computation and write:

[00 p.(t,y)A(u(y.t))dy = [pIyIAuIIIdy

—00

Using the Cauchy-Schwarz inequality,

[fI0(y)x4u(:t/)dy[2 S [ |p(y)l2dy / lAu(y)|2dy

The Parseval identity gives

[IpIy >I2dy = [lp(€)l )l2dé < [(1+€2)l( €)=|2d€ IIpII§22

Then the pseudo-differential operator form of the generator, equation (3.16) and

Lemma 3.3.2 gives

[IAU(y)l2dy = ‘1/[[ ezy€Q( (y,€ 11(2€)d€

s cw)— )1[[IQ<y.eI 2lfi(€)l2d€dy

42

s ] M2Iy) [(1+€2)|fi(€)|2d€dy

C'llPllgrzg / M 2("i/)dv

|/\

2
Now by hypothesis of the theorem f A12(y)dy < 00, therefore since pm E H 5 ’2

Vx E IR, we have [fp(y)Au(y)dy[2 < 00. Thus we can apply the Fubini theorem for

to (3.18).

Also for the negative definite symbol Q given in (3.10) the following holds:

 

Q($,€) = R] (1 — €225 — 1:”élg)vl(x,dv) — 2'8ka
0

2
If we use B(x) 2 a(x) + Cyb(x) + f vl—iggV1(x,dv) then
R0

can ——- f (1 — (2‘22 — Wat/10:, du) — it [a(x) + Cub($)]
R0

we write

Q(x,§) 2 [(1 — eTivé — ivfi) V1 (x, dv) — iC(x)€ (3.19a)
IRo

where C(x) = [a(x) +Cyb(x)].

If we use B(x) 2 a(x) — f(,/b(x) — f 1%?” (x,dv) then
IRio

43

00.35) = [R O(1 — e_iv€ )V1 (x, dv) — 23g [a(x) — 1913(1)]
we write

Q(x, ) = [R (l—e—i'U€)V1(x,dv)—iH(x)€ (3.1%)
0

In order to get a closed form of the forward equation, we need to assume V1 has
the form 1/1(1L‘,dy) 2 h(x)u(dy), where h is real valued function and ,u is a measure

on IR. Using the pseudo differential representation of A and Fubini theorem we get:

Z pIxIAquIda: = —<2w)—1/2]O[Oe22€c2(w, )&(€)p(w)d€dcv
= 7[_(2..)-1 [0 70eixie—i2’5c2e,tIpIxIdtdx]u(x’Idcc’
- pm...) I...

where Ip(x’)2 [—(27r)_1/ feixée—ixiéQw,€)p(x)d€dx[ (3.21)

—00 —OO

44

Using Q(x,§) from (3.19a)

:_(27.)-17O 70232225222 L] (1 — 23-222 435) V1(x,dv) — iC(x)€[p(x)d€dx

00 00
where 11 2 —(2rr)”1/—:/‘/]‘R eixéeTixlg(l—e—iv€—iv€)p(x)V1(x,dv)d£dx
0
oo 00

I2 = (2701 [ [e 2229 22’2IGI @thth

—00

For the first part we have,

11 2 —(27r)—1 / / feixge—i$,€(1—e—iy€ —iy€)p(x)V1(x,dy)d§dx

: —(27r)—1 70 70‘/-eim€e_iflc"$(1—e—iyé —iy§)p(x)h(x)p(dy)d§dx

—OO o—OOORO

= —(27r)1/2[ ] e-22’1—e2( 222—212)F (tr-h)(-€))u(dy)d€

: _(27r)—1/2 70 [1222,20 —e225 +iy€)F((P'h)(€))/2(dy)d€

= —()271)1/2]/ 2e2’2[-F(p h)( —Ip h)(£+y)+y(p h)’(€)]d€u(dy)

R0 -00

45

= — [ F—1 o F[(p - h)(:v’) — (p- hIIx’ + y) + yIp- h)’(rv’)]2(dy)

R0
= [ [Ip- hIIrc’ + l!) — (p - h)(rc’) — yIp~ h)’($’)]u(dv)
R0
= -/[(p h)(x —y)—(p-h)(a:’)+y(p h)’(2:’)]u(d(-y))
R0

Thus, in case we use the negative definite symbol from (3.19a), we can write

 

110(33') = - [Ra [(P' h)(:v' - y) - (ID- h)(23') + 90" h)'(rv’)] - (G - P)'($’)[
(3.22a)
Using Q(x,§) of the from (3.19b) in (3.22) Ip(x’) 2 13 + I4. Where
I3 2 —(27r)_1 edge—”TU — e_iv€)p(x)V1(x,dv)d€dx
1.1.1.

46

14 = (271)—1 f ] e222e-22'2IH(grandam.

—CXD —00

Here for the first part we have,

13 = —I2vrI‘1/2 [ [ e‘2‘2'2I1—6‘222)F(Ip-hII—tI)II(dyId€

: _(2..)—1/2 ] [.2260_e222)F(Ip-hII£I)IIIdyIdt

= —(22)‘1/2 ] ] e22'2F[(p-hII€I—Ip-hII£+yI]d52IdyI
R0—00

= _ ] 12-1 o F[(p - hIIx’I — (p. h)(:v' + yI]IIIdyI

= [ [Ip - h)(:v’ + y) — (p- h)(:v’)] #(dy)

For the second integration term,

I4 2 (2rr)—1/ feTixiéeix£i£H(x)p(x)d§dx

—OO —00
OO

: (2704/2 [ 2‘22'225F[(p-HII-€I]d§

—OO
00

= I2vrI—1/2 ] e22'2’F[Ip-HI’I€’I]da’

: F‘10F(p‘H)'(IL") = (P-H)’($')

47

Therefore, in case we use the negative definite symbol from (3.19b), we can write:

112(21): - f [(P ' h)(:v' - y) - (P- h)(~1")]u(d(-y)) - (H - p)'(:v’) (32%)

R0

Then from (3.20) and (3.18) we have,

H(y. t)1pa:(t. y)dtdy

3'\.8

a
u(y, t)a—tpx(t, y)dtdy

a
119.202) + Emmy) u(31.022612

II II
O\.8 8\8 O'\8

f—_'l

I
8

l I
0

Note that, this is true for any arbitrary choice of twice differentiable bounded u.O(-) E
L2(IR). Hence we must have [Ipx(t, y)+%p$(t, y)] 2 0, thus gzpdt, y) 2 —Ipx(t, y).

Then, combining (3.22a) and (3.22b), the forward equation is given as follows:

52pm, y) = [ [(px - hIIs, y — r) — (p... - hIIs, y) + ((122: - hI’Is, y)]u(d(-r))
IRo
--8-(px-G)(s,y) (3232)
2y

where (pa; - h)(s, y) 2 h(y)pa;(s,y) and C(y) 2 a(y) + CVb(y) ; in case CV < 00.

Alternatively,

48

gnaw): [[(px-h)(s,y—r)-(pm-h)(8.y)]u(d(-7‘))- 0%(px-H)(s,v)
1R20

(3.23b)

where (pa; - h)(s,y) 2 h(y)pg;(s,y) and H(y) 2 a(y) —- f(,/b(y) ; in case Ky < 00.

That concludes the theorem. Cl

Remark: The forward equation theorem can be used to solve an interesting
analytical problem. Let 8’ (IR) be the space of tempered distribution which is the
dual space of 8(IR). If we assume u(y, t) in S(IR), then Au(y, t) 2 —Q(y, D)u belongs
to 8 (IR) The solution of the forward equation actually produces an element in 8’ (IR),
which is the solution of the adjoint operator (forward operator) given by the transition

function.

49

Chapter 4

Application to a Special Case :
Diffusion Driven by an a-stable

Lévy Process

In this section we shall discuss a special case of SDE driven by a pure jump Lévy
process, viz. an SDE driven by a stable Lévy process. Generally, the fractional
advection dispersion or fADE is used to model a variety of anomalous diffusion
processes, where observation shows that the plume spreads away from its center of
mass faster than (t1/2) scaling implied by the Brownian motion model. This is called
super diffusion. A diffusion equation driven by an a-Stable Lévy noise can be applied
to these situations. Here we show that the transition density of the solution process
of a stochastic differential equation driven by an a-Stable Lévy process solves the

50

fADE equation.

Definition 4.0.4. A random variable X is said to have a stable distribution with
index of stability a, scale parameter a, skewness parameter S and the shift parameter

,u if its characteristic function has the following form:

exp{—oa[0[a(1—if3(sign 0) tan(%)) + i116}, if a 2 1

E[exp i6X] 2
exp{—o|0[(1+iI3-;‘);(sign 0) In |0|) + ipd}, if a 2 1

where0<ozg2,020,—1§,8$1,pEIRand

1 ,if ,0>0,
sign02 0 ,if ,0:0,
—1,if,0<0.

We write X ~ Sa(o, 8,11).

Definition 4.0.5. {X t} is a-Stable Le’vy Process if
1. X(0)2 0 as.
2. X has independent increments.

3. X(t)-X(s)~ Sa((t — s)1/a,fi,0) ,‘for0 g s < t < 00 and —1_<_ fl 3 1.

4.1 Existence and Uniqueness

Consider the SDE
dYt 2 a(Yt)dt + b(Yt)dXt (4.1)

51

here Xt is a standard, centered stable Lévy process with the index of stability 0,
(0 < a < 2) and the skewness parameter (3, (—1 g B S 1). That is, {Xt} is a Lévy
process with (Xt — X3) ~ 501 ((t — s)1/a,(3,0).

Since Xt is a pure jump Lévy process, from Theorem 2.3.1 and from section 3.1

we get the following:

Proposition 4.1.1. Suppose the coefficient functions a and b satisfy the growth

condition and Lipschitz condition as in section 3, i.e

(A) Growth condition: there exists constant C > 0 such that V y in IR
IaIyII2 + InyII2 s c(1+ IyI2)
(B) Lipschitz condition: there exists constant C’ > 0 such that V y1, y2 in IR,

la(yI) — a(y2ll2 + Ib(v1)— b(22H2 S 0' (Ii/1 - y2l2)

Then there exists a unique stochastic process {Yt} that satisfies the stochastic differ-

ential equation (4.1). Also, {Yt} is a time homogeneous Markov process.

52

 

4.2 Infinitesimal Generator, Backward and For-

ward Equations

Since {Yt} is a Markov process we can get the corresponding infinitesimal generator
and associated forward and backward equation. For an SDE driven by an a-stable
Lévy process, the infinitesimal generator of the solution process can be expressed in
terms of fractional derivatives of order a. This forward equation given in terms of
fractional derivatives of order a is used in hydrology to model ground water flows.

First let us define the fractional derivative of order a.

Definition 4.2.1. The fractional derivative of order a for a function f is derived by

solving inverse Fourier transform. Let g(x) 2 flaflx), then g(g) 2 (ii§)af(§).

Hence, g(x) :2 F—1[(ii€)af(§)].

Proposition 4.2.1. (see [3] for details.)

The fractional derivative of order a for a function f can be expressed as follows:

(a) For0<a< 1,

 

0
as 0 d
8(—$)a f(x) : m—{O (f(SU — U) — “livm (4.3)

(b)For1<a<2,

 

('30 dv
axaffi): —'—:'(%()—f/f(x—v ()+ vf’(x))lv|1+a

 

 

0
(90‘ _ a(a — 1) ,
a(_$)a f(ii?) — Era—[O (f(iF " U) - f(l‘) + Uf($))[’U[1+a

(4.4)

(4.5)

Now we return to SDE driven by a-Stable Lévy process. We shall use the following

theorem to get the Lévy type stochastic integration representation of {Yt}.

Theorem 4.2.1. (Theorem 3.12.2: [22]).

Let N be a Poisson random measure defined on [0, 00) x IRO with intensity measure

(1 +fi)dsW, if u>0,

(1‘5)d3m1%3 , if u <0.
du
lull+a

n(ds, du) 2 E[N(ds, du)] 2

2 Ifl(u)ds

 

(1+fi), if u>0,
Where [g(u) 2

(l—S), if u<0.

Now if we set 3 to be the skewness parameter of an a-Stable Le’vy process {X t}; then,

for a random function f : [0, 00) X D ——> IR,

54

(i) ForO < a <1:

[0t de3 .2. Ca f()t [no f(s)uN(ds,du)

(ii) For 1 < a < 2:

[0t fax, g Ca x (If?) ([01 /(—6,6)2 f(s)u q(ds,du))

where constant Ca is defined as follows:

(C )a (2a_1(F(1—a))cos%—f) , if 0<a<1,

_ —l
(2F2 a)(—coslrQC—')) , if 1<a<2.

a a—l)

Using the above two theorems we can derive the precise infinitesimal generator
of the solution process {Yt} of the SDE (4.1). For rest of this chapter we shall use

the following notation:

Ib(°)|a Iif b(-)>0.

—lb(-)la , z'f b(')<0.

b2I-I =

Theorem 4.2.2. Consider a stochastic difierential equation driven by an a-Stable
Le’vy process as in (4.1). Suppose the solution process {Yt} exists. Then the infinites-

55

imal generator of {Yt} is given by: if0 < a < 1,

Af(y) = a(y)f’(y) + [I1—2I(2cos(1,3))‘1b2IyIdd—a—OIIIII

+ (1+ )3) (2 cos (£22))—1ba(y)d(i:)af(y)[ (4.6a)

 

if1<oz<2,

Af(y) = a(y)f’(y) + [I1—4I(—2cos(1§))’1b WI :ny

+ (1+ 5) (—2cos (%)) 71b2(y)a,g)—df(y)] (4.6b)

Proof. Let us define a Poisson random measure N as in Theorem 4.2.1. Then,

for0<a<1

 

t t
O 0

t t
: Y0 + /a(YS)ds + Ca//D(Y3)UN(d3,dU)
0 0 R0

Let us define a Poisson random measure Na(-, ) as Na(s,u) 2 N(s, Cau). Let V be

the intensity measure of Na. Then a change of variable gives

du
[u[1+a

 

V(ds,du) = n(ds,d(Cau)) = 13(2 WdSICQ) — dSVa(du), (say)

56

where Va(du) = 13(u)(Ca)al—qul%r3 ; Ifi(-) defined as in Theorem 4.2.1. Further, we
define the compensated Poisson random measure qa by qa(-, ) : NO,(~, ~) — 17(-, o).

Thus a change of variable gives:

t t
Yt :- YO + f0.(Y3)dS + f/b(Y3) )UNa(dS,dU)
0 0R R0
t t
: Y0 + /a(Y3) )ds + / b()Y3 )uqa(ds, du)
O 0 O<|u|<1
t t
+ / f b ()Ys UNa( d3, (171) + /( b()Y3 )Ul/( (d8 ,dU)
0 |u|21 0 0<|u |<1
t
2 Y0 + /a() (“8 d3 + / /( ()()YS UdSI/a( (dU)
0 0<|u |<1

t
b((Y3)Uqa d3 ,dU) + f b ()Ys UNQ( d3 ,dU)
0 |u|21

t
+/
0

Let us denote by d, the function: a(y) = a(y) +b(y) f Ulla(du) . Then we have:
0<[u[<1

0<|u|<1

t t
0 0

0<|u|<1 IuIZI

This is of the same form as the equation given in (3.2). Also, we have function EL
in place of a and the Poisson random measure NO. From Theorem 3.2.2 using the
infinitesimal generator as in (3.3b), we can derive the infinitesimal generator in case

of an a-stable Lévy process as follows:

57

Any) = f’(y)(&(y) — mm) + / {f(y + b(y)x) — f(y)}l/a(drr)
IR0

Here, K Va 2 f xua(da:). Thus,
O<|x|<1

Af(y) = f’<y>[a(y>+b<y) / wade—be) / wade]

0<|u|<1 0<|x|<1

+ / {f(y + b(y)x) — f(y)}ua(drc)

IR0

= a(y)f’(y)+ f{f(y+b(y>x) —f(y)}ua(da:)
R0

= a(yn’o) + (can / {f(y + b(y)x) — f(y)}15(rv)l;fifi—a

R0

= a(y)f’(y) + (cadre) [My — v) — f(y)}Ifi(—v) lvl‘f‘ia

R

 

0
= a(y)f’(y)+ +(C )abaty) (1— (3)/{11 (y— u )—f(y)}l—v-— If“;

0
+(Ca)aba(y)(1+fl) /{f(y—v)— f(y)}lv—— 3:0

P(oz — 1) da
0 dyaf(y)]

P(a - 1) da

 

= a(y)f'(y) +(Ca)aba(y)(1 — m[
+(Ca>aba(y)(1 + a)[

 

Hence, for O < oz < 1, the infinitesimal generator of the solution process for SDE in

58

(4.1) can be written as:

Any) ——— a(y)f’(y) + (<1—a)(2cos(%))“1ba(y>};,-f(y>

 

 

7ra -1 a do .
+ (1+fi)(2COS(—2-)) b (y)d(_y)af(y)l (4-8)
for 1 < oz < 2: Again from Theorem 4.2.1
t t
0 0
t t
0 0(— 6,6)C

Using a similar change of variable as in previous case,

t
Yt : Y0+ /a(Y3)ds+lim/ / b(Y3)uqa(ds,du)
0 0 (— 6,6)C

t
+/ b(Ys)uNa( (ds, du)—- / b (Y3) )u17( (ds ,du)
0
O

lul21

—|/ b(Ys) )u1/(ds ,du)
UIIZ

t t
+/ /( b()YS )uqa( (ds, du) + / b (Y3) uNa( ds ,du)
0 0<[u[<1 0 |u|21

59

Let us denote by EL, the function, a(y) : a(y) — b(y) flul>1 uVa(du). Then, we have:

t t
0 0

0<[u[<1 |u|21
(4 9)

which is of the same form as the equation given in (3.2). Also we have the function
6 in place of a and the Poisson random measure Na. Mom Theorem 3.2.2 using the
infinitesimal generator as in (3.3a), we can derive the infinitesimal generator in case

of a-Stable Lévy process as follows:

Any) = my) (a(y) + Guam) + / {f(y + b(y)x) — f(y) — f’(y)b('y)x}I/a(d:v)
130

Here, CV0, = f xVa(d:1:). Therefore,
|svl21

Af(y) = f'(y)(a(y)-b(y) / uua(dU)+Cuab(z/))

|u|_>_1

+ f {f (y + b(y)x) - f (y) - f'(y)b(y)x}ua(dx)

 

R0
= a(y) () (Ca)“/{f(y+b(y>$l‘le—f'(y)b(y)$}’fi(xllxfi:a
IR0
: a(y)f’(y) + (Comm) / {f(y — v) — f(y) + f’(y)v}1ra<-v>,,—fi%
1RD

60

OO

 

 

 

= a(y)f’(y) + (Ca)aba(y>(1— a) f{f(y — v) — f(y) + f’bMfi’j—a
0
0 d
+<Ca)aba<y>(1+m / {f(y—v) —f(y)+f’(y)v}lvl1:a
= a(yn’o) + (Ca)aba(y)(1—m[:E:‘_j’( diam]
+(Ca>aba(y)<1 + m [2):ng d(fy)af(y>]

Hence, in case we have 1 < oz < 2, the infinitesimal generator for SDE in (4.1) can

be written as:

 

Combining Equation (4.8) and Equation (4.10) we get the infinitesimal generator in

(4.6), and the theorem is proved. Cl

The Backward equation :

 

The backward equation can be obtained using Theorem 3.2.3. Let A be the infinites-
irn a1 generator for the solution process {Yt} of SDE driven by a-Stable Lévy process
as in (4.1). Let f 6 03m),

D813 ne, u(y,t) : Ey [f(Yt)] . Then,%‘ti : A(u) . That is:

61

fiO<a<1

 

 

 

3:5,: t) = a(y)U'(y,t) + [(1— m (2cos<%>)’1ba<y);—:u<y, t)
+(1+ fi)(2 005C?» -lba(y)d(fy)au(y,t)] (4.11a)
fil<a<2
9%? = a(y)U'(y,t) + [(1 — m( — 2cos<329))’lba<y)g5u(y,t>
+(1+ fl) ( — 2cos(%))_lba(y) d(iy)au(y, 15)] (4.11b)

The Forward equation :

 

Theorem 3.3.2 can be used to obtain the forward equation. Let {Yt} be the solution
process for the SDE in (4.1). Let us further assume that there exists a transition
probability density for Yt and let px(t, y) be the transition p.d.f of Yt, given Y0 = as.
We have already stated that the Lévy measure for the a-Stable Lévy process {X t}
is given by law?» = Igoxcaalgfisz.

Note that, the measure 111(1‘, u) in Theorem 3.3.2 is the measure derived from the
Lévy measure Va(dy) by change of variable —u = b($)y. So in this case the change
of variable leads to 111(13, u) : 13(u)(Ca)a(b(x))a|—Jfiy+—a.

Now we see V1 is of the form 111(1), dy) = h(:c),u(dy), where h(:1:) : ba(:1:) and p E Va.

Thus, if condition (3.14) holds, then the transition probability density function of

62

Yt satisfies the forward equation given in Theorem 3.3.2. This leads us to the next

theorem.

Theorem 4.2.3. ( The forward equation for an a-stable Lévy diffusion)
Consider a stochastic differential equation driven by an a-stable Le’vy process (01 yé 1,
O < a < 2) as in (4.1), such that the solution process Yt exists and is unique. Then
if the coefi‘icient functions a and b satisfy assumption (3.14), and if there emists
a transition probability density function p1;(s,y) of the solution process {Y3} given
Y0 = :13, then the following forward equation holds:

if0<oz<1,

gpx(s,y) = [(1 + fi)(2005(%€))—1g;a(ba(y)px(s,y))

 

+<1 — m (2cos ("—2‘3'-))"1 (“f/)0. (Maw, 20)] — i:- [a(y)px(s,y)]<4.12a>

fil<a<z

7m -1 do‘

gain) = ((1+s)(—2cos(7)) @(buymaayn

 

+<1 — m (—2 (”—2‘J’-))_1 (“$6. (ba(y)px(s, 30)] — flame, y)](4.12b)

Proof. Case I : 0 < a < 1: In this case

(1
Kl/a Z (Ca)a / $IB($)-|;;|1$m Z 2(Ca)a B

 

<00.
l—a

63

Therefore we shall use the forward equation from (3.3b), i.e,

Eighty) = / [(px - may — r) - (pm - h)(8,y)]V(d(-T)) — Egon: - may)
R0

where (Pa: - h)(w) = h(9lP$(3ay) and H (y) = a(y) - Kuabty).

Note, in this case H(y) : a(y) —— KVOb(y) : a(y). Thus, the forward equation:

a a a a
7mm): / [(px-b )(s.y—r>—(pxob )(s,y>]ua(d(—r))—gummy)

(‘3
R0
or a 01 d7 0
= (Ca) R/[(Px'b )(s,y—r)—(px-b )(say)]1p(-T)W—a—ybixoaflsay)
0

: (1+ mean / [(px - may — r) — (pa: - were] W
0

0
+(1— WOO)“ / [(px -ba)(s,y — r) — (P2: - ba)(s,y)lI—T|1—+(;

2 [(1+ fl) (2 cos (£23) )—1ga(ba(y)px(8,y))

 

+(1 — fl) (2 C05 (7129') )_1d(i:)a (ba(ylpx(3ay))] " 2%" [a(y)px(8, 31)]

Case II : 1 < a < 2: In this case

 

 

CV0 = / mVa(d:r) : 2(Ca)a
lxlzl

01—1

64

Therefore we shall use the forward equation from (3.3a) here, i.e,

grass!) = / [(px-th-r,s)—(pa:-h)(s,y)+r(px-h)'(y,8)]M(d(-7'))-%(Px'cl(y,S)
R0

where (Pa: - h)(s, y) = h(v)px(s,y) and C(31) = a(y) + Cuab(y)-

In this case C(y) = a(y) + Cuab(y) : a(y) . Thus, the forward equation is given by:

gm“): f[<px-ba>(y—r,s)—(mine, snaps. We s>]ua(d(— —v~>)

R0
— %(px-a)(y,8)
a a o a I dr
= (Ca) / [ox-b )(s,y—r)—(px-b )(s.y>+r<px-b mom—om
R0
— 3%(p33 (”(3,111)
= <1+fi><ca)af[(px-ba)(s,y—r)— (px bows y)+r(px bans y)]fia
0
0 d
+(1—6)<Ca)a / [(pm we y—r)— (Pm ms y)+r<px bans y)]Ir—T—lia

8

_ 5%(p1' ' a)(39 y)

= [(1+m( 20041;» 5;: (bad (mew)

u—m<— mes)

 

:Iab"(( ()p:c(s y))]- 5%[a(y)px(s,y)]

We combine the two cases to get the forward equation of the form (4.12). This

concludes the proof. 1:]

65

Remark : The above fractional derivative form representation of the forward
equation is just a special representation in this case. This form agrees with the
space-fractional advection-dispersion equation used in hydrology. The main forward

equation form derived as in (3.3a) or (3.3b) can be used for analytical purposes.

66

BIBLIOGRAPHY

[1] Applebaum, D. : Lévy Processes and Stochastic Calculus, Cambridge Univer-
sity Press, (2004).

[2] Bass, R.F. : Uniqueness in law for pure jump Markov processes, Probability
Theory and Related Fields, 79, p:271-287, (1988).

[3] Baeumer, B., Meerschaert, M.M. : Stochastic Solutions for Fractional Cauchy
Problems, Fractional Calculus and Applied Analysis, Vol. 4, No. 4, (2001).

[4] Bear, J. : Dynamics of Fluids in Porous Media, Dover, Mineola, N. Y., 1972.

[5] Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M. : The Fractional Order
Governing Equation of Lévy Motion, Water Resources Research, Vol. 36, No.
6, June (2000).

[6] Bhattacharya, R.N., Gupta, V.K and Sposito, G. : On The Stochastic Founda-
tion of the Theory of Water Flow Through Unsaturated Soil, Water Resources
Research, Vol. 12, No. 3, June (1976).

[7] Bhattacharya, RN, and Gupta, V.K. : A Theoretical Explanation of Solute
Dispersion in Saturated Porous Media at the Darcy Scale, Water Resour. Res.,
19(4), 938-944, (1983). '

[8] Billingsley, P. : Convergence of Probability Measures, John Wiley & Sons,
(1968).

[9] Bottcher, B. and Schilling, R.L : Approximation of Feller Processes by Markov
Chains with Lévy Increments, Stochastics and Dynamics, Vol: 9 Issue: 1, Page:
71 - 80, (2009).

67

[10] Cartea, A., and D. del-Castillo-Negrete : Fluid limit of the continuous-time
random walk with general Lévy jump distribution functions, Phys. Rev. E, 76,
041105, (2007).

[11] Chakraborty, P. : A Stochastic Differential Equation Model with Jumps for
Hactional Advection and Dispersion, to be appeared in The Journal of Statis-
tical Physics.

[Available in https:\\ www.stt.msu.edu\ ~chakrab.]

[12] Chakraborty, P., Meerschaert M.M and Lim, C.Y. : Parameter Estimation for
Ffactional Transport: A Particle Tracking Approach, to be appeared in Water
Resources Research.

[Available in https:\\ www.stt.msu.edu\ ~chakrab.]

[13] Durret, R. : Probability: Theory and Example, Third Edition, Thompson
Books/Cole.

[14] Fetter, C.W. : Applied Hydrogeology, 4th ed., Prentice Hall, New York, 598
pp (2001).

[15] Gupta, V.K., Sposito, G. and Bhattacharya, R.N. : Toward an Analytical
Theory of Water Flow Through Inhomogeneous Porous Media, Water Resour.
Res., 13(1), 208-210, (1977).

[16] Hoh, W.A. : A symbolic calculus for pseudo differential operators generating
Feller semigroups, Osaka Journal of Mathematics, 35, p:798—820, (1998).

[17] Ikeda, N. and Watanabe, S. : Stochastic Differential Equations and Diffusion
Procws, North-Holland Publishing Company, (1981).

[18] Jacob, N. : Pseudo-differential operators and Markov process (Vol I and II),
Imperial College Press, (2002).

[19] Jacob, N. and Schilling, R.L : Levy-type processes and pseudo-differential oper-
ators, Lévy Processes - Theory and Applications (ed. S. Resnick, O. Barndorff-
Nielsen and Th. Mikosch) Birkhuser Verlag, 2001, 139-168.

[20] Rosiriski, J. : Tempering stable processes, Stochastic Process. Appli., 117,
677-707, (2007).

68

[21] Rudin, W. : Principle of Mathematical Analysis, McGraw Hill, Third Edition.

[22] Samorodnitsky, G. and Taqqu, M. S. : Stable Non-Gaussian Random Processes,
Stochastic Models with Infinite Variance, Chapman & Hall, New York, (1994).

[23] Sato, K.I. : Lévy Processes and Infinitely Divisible Distributions, Cambridge
University Press, (2004).

[24] Skorokhod, A.V. : Studies in the theory of random processes, Addison-Wesley
Publishing Company,Inc., (1965).

[25] Zhang, Y., Benson, D., Meerschaert, M.M. and Scheffler, H.P : On Using Ran-
dom Walks to Solve the SpacaFractional Advection-Dispersion Equation, J our-
nal of Statistical Physics, Vol. 123, No.1, April (2006).

[26] Zhang, Y., Benson, D., Meerschaert, M.M., LaBolle, E. and Scheffler, H.P :
Random walk approximation of fractional—order multiscaling anomalous diffu-
sion, Physical Review E, 74, No.2, 706-715 (2006).

[27] Zhang, Y., Benson, D., Meerschaert, M.M. and LaBolle, E. : Space-fractional
advection-dispersion equations with variable parameters: Diverse formulas, nu-
merical solutions, and application to the MADE-site data, Water Resources

Research, Vol. 43, p. W05439, doi:10.1029/2006WR004912. (2007).

69

Ill“

llllllllllllllllll

03062 972

3
9
2
1
3

"
"II
I"
H
II]
Am
ml]
Hl