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On the non-degenerate parabolic Kolmogorov integro-differential equation and its applications
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On the non-degenerate parabolic Kolmogorov integro-differential equation and its applications
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Content
ON THE NON-DEGENERATE PARABOLIC
KOLMOGOROV INTEGRO-DIFFERENTIAL EQUATION
AND ITS APPLICATIONS
by
Fanhui Xu
A dissertation presented to the faculty of the USC GRADUATE SCHOOL at the
University of Southern California
in partial fulllment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in MATHEMATICS
in the city of Los Angeles in May, 2019.
Copyright ' 2019
All Rights Reserved
Dedicated to my beloved mother, Yu’e Li.
ii
Acknowledgments
This thesis is impossible without the help from my advisor Prof. Remigijus Mikulevi cius.
First and foremost, I would like to thank him for teaching me the fundamentals of stochastic
analysis and pointing out these two wonderful projects to work on. He is a great lecturer
and the semester when I took Real Analysis with him was one of the happiest periods in
my graduate student life. I am indebted to Prof. Igor Kukavica for his role in my on-going
scholarship. The level of attention and support he gives to every student is unprecedented
in terms of what I have seen in other Ph.D. programs across the country. I can never thank
him enough for acknowledging my potential in mathematics from the very beginning and
for boosting my condence at crucial moments of my career. The impact he has had on me
is profound and will guide me to become a good educator and mathematician.
My gratitude also goes to Prof. Peter Baxendale, who is incredibly generous about
providing insights behind the research project and helping young researchers grow. The
tremendous amount of time he spent on my presentation rehearsals was such a tranquilizer
for me during the period of job application. Many heartfelt thanks to Prof. Nabil Ziane,
for his generous help in my career. He is a wonderful mathematician and professor, who
makes math simple, charming, and fun without losing any of its complexity and rigor. The
answers he provided in our conversation regarding \how to learn from a talk" and \why do
we conduct research in this way", together with his intuitive way of teaching, have become
my academic philosophy.
I would also like to thank Prof. Roger Ghanem for being my committee professor and
Prof. Sergey Lototsky for oering multiple presentation opportunities in the probability
iii
seminar and asking inspiring questions. Many thanks to Prof. Susan Montgomery for regu-
larly checking on my research progress and always being caring and supportive. Besides, I
would like to genuinely thank Prof. Cymra Haskell and Prof. Juhi Jang for their remarkable
in
uence on the women in math community.
I would not make this far had I not been surrounded by the support and encouragement
of my amazing friends. Nicolle Sandoval Gonz alez, it is a blessing to have you in my life.
Accidentally or non-accidentally, we got on the same \roller coaster", going through ups
and downs all together. I can not imagine screaming at those downs without you screaming
next to me. Thank you for being a constant presence around and being an intimate friend
who answers my call at 3am. Bahar Acu, your intelligence and courage to overcome any
diculty and your will to help others have been an inspiration for me all these years. Thank
you very much for giving me a belief and a goal. Ujan Gangopadhyay, you are too non-
trivial to be described by a trivial adjective. Your reliable and pleasant show-ups at Dulce
together with your objective comments on my struggles brought me so much peace and
calm. Guher Camliyurt, the encouraging words you said mean a ton to me and you said a
ton of them. Think about the value of this exponential. You dene what a perfect friend is.
Chinmoy Bhattacharjee, my graduate student life would be just black and white if it was
not for your smiley face and solo singing in our oce. Melike Sirlanci, thank you for those
conversations at Illy Cafe. You were never absent whenever I needed. Ozlem Ejder, your
cheerful spirit shed light on my carer path, thank you for putting so much eort into my rst
talk at Charlotte's Web. Hyun-jung Kim, it feels so great to have a research accompany like
you, wise and modest. Moumanti Podder, thank you for thinking of me as a true friend the
rst time we met and thank you for your generous support in the job-application season.
Last but not least, many thanks to Ting Yao, Helen Kim, Can Ozan Oguz, Ezgi Kantarci,
Cindy Blois, Julian Aronowitz, Zhanerke Temirgali, Eilidh McKemmie, John Rahmani,
Shuang Li, and Xiaojing Xing.
iv
Table of Contents
Dedication ii
Acknowledgments iii
Abstract 1
Notation 2
Part I: Numerical Analysis for SDEs Driven by L´ evy Jump Processes 4
Chapter 1: Introduction 5
1.1 Stochastic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Model Description and Main Results . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 2: Auxiliary Estimates 14
2.1 Stochastic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Backward Kolmogorov Equations in H older Classes . . . . . . . . . . . . . . 18
2.3 Estimates of Stochastic Integrals and Driving Processes . . . . . . . . . . . 22
Chapter 3: Proof of Main Results 33
3.1 Lipschitz Drift and Non-truncated Noise . . . . . . . . . . . . . . . . . . . . 33
3.2 Lipschitz Drift and Truncated Noise . . . . . . . . . . . . . . . . . . . . . . 37
3.3 H older Drift and Non-truncated Noise . . . . . . . . . . . . . . . . . . . . . 39
3.4 H older Drift and Truncated Noise . . . . . . . . . . . . . . . . . . . . . . . . 48
Part II: The Cauchy Problem for Parabolic Non-degenerate Kolmogorov
Equations in Generalized H¨ older Spaces 55
Chapter 4: Introduction 56
4.1 Motivation and Problem Description . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Properties of O-RV Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 5: Norm Equivalence 69
5.1 Equivalent Norms on
8
pR
q . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Extension of Norm Equivalence to
r
88
pR
q . . . . . . . . . . . . . . . . . 86
v
Chapter 6: Proof of Main Theorem 90
6.1 H older Estimates of the Smooth Solution . . . . . . . . . . . . . . . . . . . 91
6.2 General H older Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Bibliography 99
vi
Abstract
This dissertation consists of two parts. In Part I, a stochastic dierential equation (SDE)
driven by an -stable process with Pr1, 2q is considered. It is assumed that this SDE has
a Lipschitz jump coecient and a -H older continuous drift with Pp0, 1s. The existence
and uniqueness of a pathwise solution is proved for ¡ 1 {2 by showing that it is the
-limit of Euler approximations. When P p0, 1q, the
-error (rate of convergence) is
obtained for non-degenerate truncated and non-truncated driving processes. When 1,
the rate of convergence is derived for possibly degenerate truncated and non-truncated
driving processes. In the case where 1, regularity properties of the solution to a
Kolmogorov integro-dierential equation are used to overcome the diculty brought by the
rough jump coecient.
In Part II, a Cauchy problem for a non-degenerate parabolic Kolmogorov integro-
dierential equation is investigated. The primary operator of this equation is an innites-
imal generator of a L evy process which has a radially O-regularly varying L evy measure.
Function spaces of generalized H older smoothness are dened using the L evy measure. Exis-
tence and uniqueness of a strong solution, together with its space and time regularity are
established in generalized H older spaces.
1
Notation
Basic Notation
R
: The -dimensional Euclidean space.
R
0
: R
zt0u.
R
: All positive real numbers.
ℬpR
q: Borel -algebra on R
.
||: Lebesgue measure of , PℬpR
q.
1
: The unit sphere in R
.
N :t0, 1, 2, 3,...,u.
N
Nzt0u.
:r0,s R
.
1
:r0, 1s R
.
: The -identity matrix.
p,q :
°
1
, ,P R
.
|| :p,q
1{2
, P R
.
r s: The greatest integer that is less than or equal to .
t u : r s.
|| : sup
PR
,||1
||, where is a matrix.
}} : sup
PR
|pq|, where is a matrix function on R
.
|∇|
8
: The Lipschitz constant of .
p,...,q: A constant depending only on quantities appearing in the parentheses.
2
Notation for Functions
For a function on
, we denote its partial derivatives by B
B{B, B
B{B
,
B
2
B
2
{B
, and denote its gradient with respect to by ∇ pB
1
,...,B
q.
| |
:B
| |
{B
1
1
...B
, where p 1
,...,
qP N
is a multi-index.
Meanwhile, we write
||
0
sup
,
|p,q|,
rs
sup
,,ℎ0
|p,ℎqp,q|
|ℎ|
if Pp0, 1s.
Notation for Function Spaces
We use
8
pR
q for the set of innitely dierentiable bounded functions on R
whose
derivative of arbitrary order is also bounded, and
pR
q, P N for the class of -times
continuously dierentiable functions.
For r st u¡ 0,
p
q denotes the space of measurable functions on
such
that the norm
||
¸
| |¤r s
|
|
0
¸
| |r s
r
s
t u
8,
and
pR
q denotes the corresponding space of functions on R
.
We denote the Schwartz space of rapidly decreasing functions on R
by pR
q and the
space of continuous functionals onpR
q by
1
pR
q, i.e. the space of tempered distributions.
We adopt the normalized denition for the Fourier and the inverse Fourier transforms
for functions in pR
q, i.e.,
ℱ p q p p q :
»
2 pq,
ℱ
1
pq pq :
»
2 p q, PpR
q.
Recall that Fourier transform can be extended to a bijection on
1
R
.
3
Part I
Numerical Analysis for SDEs Driven by
L´ evy Jump Processes
4
Chapter 1
Introduction
We start this chapter with some fundamentals of stochastic analysis.
1.1 Stochastic Preliminaries
Let p, ℱ, Pq be a complete probability space. A stochastic process t
u
¥0
is a
parametrized collection of random variables dened onp, ℱ, Pq and taking values in R
.
We can also writet
u
¥0
as a function-valued random variable
pq and each \value" is
called a sample path of .
Stochastic processes are mathematical models describing time evolution of various ran-
dom phenomena. The most fundamental process used for modeling continuous random
motions is the Wiener process, and for discontinuous random evolutions is the compound
Poisson process. Both of them are examples of L evy processes.
Definition 1.1.1. A stochastic processt
:¥ 0u is L´ evy (in law) if
1.
0
0 a.s.
2. For any ¥ 1 and arbitrary 0¤
0
1
, the random variables
0
,
1
0
, ,
1
are independent.
3. The distribution of
does not depend on .
4.
is stochastically continuous, i.e., for any ¡ 0,
Pp|
|¡qÝÑ 0 as ÝÑ.
Clearly, a L evy process has independent and stationary increments, which implies innite
divisibility of its distribution.
5
The well-known dening property, L evy-It^ o decomposition entails that every L evy pro-
cess in R
can be decomposed into
»
0
»
||¡1
p,q
»
0
»
||¤1
p,q, (1.1.1)
where P R
,
is a -dimensional Gaussian L evy process with a covariance matrix ,
p,q is a Poisson random measure, and p,q is the compensated Poisson measure,
i.e.,
p,qp,q Ep,q.
Definition 1.1.2. A Radon measure on R
is called a L´ evy measure if is positive
and veries
pt0uq 0,
»
R
p||
2
^ 1q pq 8. (1.1.2)
Given a L evy measure , there exists a unique Poisson random measure p,q such
that
Ep,q pq, p,qPr0,s R
,
and vice versa.
Definition 1.1.3.The distribution of each L evy process is determined by a -dimentional
vector , a positive denite matrix , and a L evy measure , andp,, q is called
the characteristic triplet of that L evy process.
Because L evy processes are innitely divisible, it suces to specify their distributions
at any xed time.
Definition 1.1.4.An -stable process is a L evy process that has an -stable distribution
at 1.
The -stable process has been extensively used in physical modeling and mathematical
nance (see [4,10]). As Ñ 2, it approaches a Brownian motion.
6
Proposition 1.1.5 ( [34], Theorem 14.3). A -dimentional L´ evy process is -stable, P
p0, 2q, if and only if it has characteristic tripletp0
,, q and there exists a finite measure
on
1
such that for any Borel set in R
,
pq
»
1
pq
»
8
0
1
pq
1 .
Remark 1.1.6. If pq is a nonnegative measurable bounded function on R
such that
p0q 0, p q pq for all ¡ 0, and
0
¤ pq¤,@ 0 for some
0
,¡ 0, and
pq pq
||
,
then there exists a unique Poisson random measure p,q on r0,s R
satisfying
Ep,q pq, and then by (1.1.1), there is a unique L evy process
»
0
»
||¡1
p,q
»
0
»
||¤1
p,q, (1.1.3)
where
p,qp,q pq,
and
$
'
'
'
'
&
'
'
'
'
%
³
||¤1
pq
||
if 0 1,
0 if 1,
³
||¡1
pq
||
if 1 2.
Equivalently, we may write
as
»
0
»
p1 pqqp,q
»
0
»
pqp,q, (1.1.4)
where pq : 1
Pp1,2q
1
1
1
||¤1
. By Proposition 1.1.5,
is an -stable process whose
characteristic triplet isp0
,, q.
7
Definition 1.1.7.An infinitesimal generator , say , for a L evy process
is the oper-
ator satisfying
pq
pq for any P
2
0
pR
q,
where
pq : Erp
qs
is the transition operator for this L evy process.
Proposition 1.1.8 ( [34], Theorem 31.5). Let t
: ¥ 0u be a -dimentional -stable
process with characteristic triplet p0
,, q. Then the infinitesimal generator of is
defined for anyP
2
0
pR
q as
pq ∇pq
»
R
pqpq∇pq1
||¤1
pq.
Remark 1.1.9. By the remark of Proposition 1.1.5, the innitesimal generator of
dened
in (1.1.4) can be written as
pq
»
R
pqpq pq∇pq
pq, (1.1.5)
where pq 1
Pp1,2q
1
1
1
||¤1
.
Remark 1.1.10.
pq is a solution to the integro-dierential equation
B
p,qp,q. (1.1.6)
In Chapter 3, we will see how this type of parabolic integro-dierential equation helps to
solve a SDE driven by the corresponding stochastic process.
8
1.2 Model Description and Main Results
Letp, ℱ, Pq be a complete probability space andFpℱ
q
Pr0,1s
be a ltration of -algebras
satisfying the usual conditions. Suppose pq is a nonnegative measurable bounded function
on R
such that p0q 0, p q pq for all ¡ 0, and
0
¤ pq¤ ,@ 0 for some
0
,¡ 0. By the discussion in last section, there exists an adapted Poisson random measure
p,q onr0, 1s R
with
Ep,q pq
||
, Pr1, 2q.
Meanwhile,
»
0
»
p1 pqqp,q
»
0
»
pqp,q
denes an -stable process, Pr1, 2q. Note the compensated Poisson random measure
p,qp,q pq
||
is a martingale measure.
In Part I, we consider the following SDE driven by
in the time intervalrs:
0
»
0
p
q
»
0
p
q
. (1.2.1)
Here, the drift coecient : R
Ñ R
is a bounded function of -H older continuity in
the whole space with P p0, 1s and pq, P R
, is a Lipschitz continuous bounded
-matrix. We assume for S(,
0
). (i) p0q 0,
0
¤ pq¤, P R
0
for some
0
,¡ 0;
(ii) p q pq for all ¡ 0, P R
0
, i.e., is a 0-homogeneous function;
(iii)
pq pq, P R
0
, if 1. (1.2.2)
9
We are going to study the Euler approximation to (1.2.1) dened as
0
»
0
pq
»
0
pq
, (1.2.3)
where
pq{ if { ¤p 1q{, 1, 2,..., 0,..., 1.
Sometimes in (1.2.1)
is replaced by its truncation
0
»
0
»
||¤1
p,q, Pr0, 1s,
i.e., the following equation and Euler approximation are considered instead,
0
»
0
p
q
»
0
p
q
0
, Pr0, 1s, (1.2.4)
and
0
»
0
pq
»
0
pq
0
, Pr0, 1s. (1.2.5)
This case will also be addressed in this thesis. It is well-known that the truncated driving
process
0
has all moments, while
only has moments up to and is not included.
It is a concern of many mathematicians that whether or not the presence of a noise term
eliminates the notorious non-uniqueness of solutions to the ordinary dierential equation
with a H older drift. The answer has been given for the white noise. But much less is
known for jump noise. In [33], the existence and uniqueness of strong solutions to (1.2.1)
was considered by assuming
, the -identity matrix, and
is a symmetric
non-degenerate -stable process with P r1, 2q, ¡ 1 {2. The pathwise uniqueness
for (1.2.1) was proved by applying Gr onwall's lemma and using the elliptic version of the
Kolmogorov equation and regularity of its solution, to represent the H older drift pq by
an expression which is \Lipschitz". This approach, \It^ o-Tanaka trick", was inspired by
considerations in [17], see the innite dimensional generalization in [11] for and
being Wiener, or a nite dimensional generalization (using parabolic backward
10
Kolmogorov equations) in [16], again with
, , and having some integrability
properties.
On the other hand, in [32] a truncated equation (1.2.4) and its Euler approximation
(1.2.5) were considered with
, 1. Using the same It^ o-Tanaka trick and assuming
that a strong solution
exists with ¡ 2, Pp0, 1q, the rate of strong convergence
was derived. It was proved in [32] that
E
sup
|
|
¤
$
'
&
'
%
1
if ¥ 2{,
{2
if 2¤ 2{.
(1.2.6)
In Part I, using It^ o-Tanaka trick again, we derive the rate of convergence of Euler
approximations for both (1.2.1) and (1.2.4). We show that, under the imposed assumptions,
,
are Cauchy sequences whose limits solve (1.2.1) and (1.2.4) respectively.
For (1.2.1), the following holds. Note that only the moments exist in this case.
Proposition 1.2.1. Let P r1, 2q, Sp,
0
q hold, P p0, 1q and ¡ 1 {2. Assume
P
pR
q, is bounded and Lipschitz, and | detpq| ¥
0
¡ 0, P R
, i.e. is
uniformly non-degenerate. Let for some
1
¡ 0,
| pq pq|¤
1
||
for all|||| 1.
Then there is a unique strong solution to (1.2.1). Moreover for each Pp0, q, there is
depending on ,,,,,, such that
E
sup
0¤¤1
|
|
¤
{ .
For (1.2.4) we derive the following statement which extends and improves the results
in [32], see (1.2.6).
11
Proposition 1.2.2. Let P r1, 2q, Sp,
0
q hold, P p0, 1q and ¡ 1 {2. Assume
P
pR
q, is bounded Lipschitz and | detpq|¥
0
¡ 0, P R
, i.e. is uniformly
non-degenerate. Let for some
1
¡ 0,
| pq pq|¤
1
||
for all|||| 1.
Then there is a unique strong solution to (1.2.4). Moreover for each Pp0,8q, there is
depending on ,,,,,, such that
E
sup
0¤¤1
|
|
¤
$
'
'
'
'
&
'
'
'
'
%
{ if 0 {,
p{ lnq
1
if {,
1
if ¡ {.
In both statements above, and are non-degenerate (Assumption Sp,
0
q holds).
On the other hand, if and are Lipschitz and bounded functions, then there exists a
unique solution to (1.2.1) (see Theorem 6.2.3, [3]) with any bounded nonnegative . For
this possibly completely degenerate case, we use direct estimates of stochastic integrals to
derive the convergence rate.
The following statement holds under the assumption that both and are Lipschitz.
Proposition 1.2.3. Let Pr1, 2q, be nonnegative bounded. Assume and are bounded
Lipschitz functions. Then
(i) For each Pp0, q, there is depending on ,,,,, such that
E
sup
0¤¤1
|
|
¤ p{ lnq
{ if 0 Pp1, 2q,
E
sup
0¤¤1
|
|
¤ r{plnq
2
s
if 0 1.
(ii) If 1, and pq pq, P R
, then there is depending on ,,,,, such that
E
sup
0¤¤1
|
|
¤p{ lnq
if 0 1.
12
We derive the following rate of convergence in all Lipschitz cases for (1.2.4).
Proposition 1.2.4. Let Pr1, 2q, be nonnegative bounded. Assume and are bounded
Lipschitz functions. Then
(i) For each Pp0,8q, there is depending on ,,,,, such that
E
sup
0¤¤1
|
|
¤
$
'
'
'
'
'
'
'
&
'
'
'
'
'
'
'
%
p{ lnq
{ if 0 Pp1, 2q,
{plnq
2
if 0 1,
{plnq
2
1
if ,
1
if ¡ (ii) If 1, and pq pq, P R
, then there is depending on ,,,,, such that
E
sup
0¤¤1
|
|
¤p{ lnq
if 0 1.
The rates above are in agreement with the subtle results obtained in [18] for (1.2.1) in
the case 1, 0, P
3
.
An obvious consequence of Proposition 1.2.3 is
Corollary 1.2.5. Let P r1, 2q, be nonnegative and bounded. Assume and are
bounded Lipschitz functions. Then
(i) there is depending on ,,,,, such that for each P
pR
q, Pp0, 1s, P
r0, 1s,
|E p
q E p
q| ¤ | |
p{ lnq
{ if Pp1, 2q,
|E p
q E p
q| ¤ | |
r{plnq
2
s
if 1.
(ii) If 1, and pq pq, P R
, then there is depending on ,,,,, such that for each P
pR
q, Pp0, 1q, Pr0, 1s,
|E p
q E p
q|¤| |
p{ lnq
.
13
Chapter 2
Auxiliary Estimates
2.1 Stochastic Tools
In this section, we state some existing inequalities in the area of stochastic analysis and use
them to conclude some results that are needed in the following sections. Let us start with
Lenglart's inequality (see [24]).
Suppose
is a nonnegative c adl ag process and
is an increasing predictable process.
We say that dominates if for any nite stopping time ,
E
¤ E
.
Lemma 2.1.1 (Corollary II, [24]). Let be dominated by . Then for every Pp0, 1q and
every stopping time ,
E
sup
¤ |
|
¤
2
1
Er
s.
Remark 2.1.2. Let :r0, 1s
R
0
Ñ R
be a ℬpR
0
q-measurable function, :
pq, Pr0, 1s, P R
. Assume that for any Pr0, 1s a.s.,
»
0
»
|
pq|
2
pq
||
8,
where is a predictable -algebra onr0, 1s
. Then
(i) (See [24])
»
0
»
pqp,q
2
, Pr0, 1s,
is dominated by
»
0
»
|
pq|
2
pq
||
, Pr0, 1s.
14
By Lemma 2.1.1, for any Pp0, 2q there is pq such that for any stopping time ,
E
sup
¤
»
0
»
pqp,q
¤E
»
0
»
|
pq|
2
pq
||
{2
.
(ii) On the other hand, for Pr1, 2s, by the BDG inequality,
E
sup
¤
»
0
»
pqp,q
¤ E
»
0
»
|
pq|
2
p,q
{2
¤ E
»
0
»
|
pq|
p,q
¤ E
»
0
»
|
pq|
||
. (2.1.1)
Remark 2.1.3. Let :r0, 1s
R
0
Ñ R
be a ℬpR
0
q-measurable function, :
pq, Pr0, 1s, P R
, such that for any Pr0, 1s a.s.,
»
0
»
|
pq| pq
||
8.
(i) Obviously,
»
0
»
pqp,q
, Pr0, 1s,
is dominated by
2
»
0
»
|
pq| pq
||
, Pr0, 1s.
By Lemma 2.1.1, for any Pp0, 1q there is pq such that for any stopping time ,
E
sup
¤
»
0
»
pqp,q
¤E
»
0
»
|
pq| pq
||
.
(ii) For Pr1, 2s, by BDG inequality, we have as in (2.1.1),
E
sup
¤
»
0
»
pqp,q
¤ E
»
0
»
|
pq|
2
p,q
{2
¤ E
»
0
»
|
pq|
||
.
15
For the sake of completeness we remind two other \general" estimates.
Lemma 2.1.4 (Lemma 4.1, [23]). (i) (Kunita’s inequality) Let :r0, 1s
R
0
Ñ R
be a ℬpR
0
q-measurable function, :
pq, P r0, 1s, P R
, such that for any
Pr0, 1s a.s.,
»
0
»
|
pq|
2
pq
||
8,
where is a predictable -algebra on r0, 1s
. Then for each ¥ 2 there is pq
such that for any stopping time ,
E
sup
¤
»
0
»
pqp,q
¤E
»
0
»
|
pq|
pq
||
E
»
0
»
|
pq|
2
pq
||
{2
.
(ii) Let :r0, 1s
R
0
Ñ R
be aℬpR
0
q-measurable function, :
pq,P
r0, 1s,P R
, such that for any Pr0, 1s a.s.,
»
0
»
|
pq| pq
||
8,
Then for each ¥ 1 there is pq such that for any stopping time ,
E
sup
¤
»
0
»
pqp,q
¤E
»
0
»
|
pq|
pq
||
E
»
0
»
|
pq| pq
||
.
Remark 2.1.5. Let :r0, 1s
R
0
Ñ R
be a ℬpR
0
q-measurable function, :
pq, Pr0, 1s,P R
, such that for any Pr0, 1s a.s.,
»
0
»
|
pq| pq
||
8.
16
(i) Since p,q-stochastic integral is a sum, a.s. for every Pp0, 1q, Pr0, 1s,
»
0
»
pqp,q
¤
»
0
»
|
pq|
p,q.
Hence for any stopping time ,
E
sup
¤
»
0
»
pqp,q
¤ E
»
0
»
|
pq|
pq
||
.
(ii) On the other hand,
³
0
³
pqp,q
, Pr0, 1s, is obviously dominated by
»
0
»
|
pq| pq
||
,Pr0, 1s.
By Lemma 2.1.1, for each Pp0, 1q, there is pq¡ 0 so that
E
sup
¤
»
0
»
pqp,q
¤E
»
0
»
|
pq| pq
||
.
We will use the statement below to derive a global estimate from a local one.
Lemma 2.1.6. Let
, Pr0, 1s, be a nonnegative c` adl` ag stochastic process,
0
0, and
¡ 0. Assume there is Pp0, 1q and , ¡ 0 such that for any 0¤ ¤ ¤ 1 with
||¤, we have
E
sup
¤¤
¤rEr
ss.
Then there is p, q so that
E
sup
0¤¤1
¤.
Proof. We partition r0, 1s into
0
subintervals of length
1
0
¤ . Let
{
0
,
0,...,
0
, and
E
sup
1
¤¤
, 1,...,
0
.
17
then,
1
¤ and
¤
1
for 2,...,
0
, and then
¤p
...q
, 1,...,
0
.
Therefore,
E
sup
0¤¤1
¤p
1
...
0
q.
2.2 Backward Kolmogorov Equations in H¨ older Classes
We will rely on some results about backward Kolmogorov equations. Recall that is
assumed to be Lipschitz continuous, which implies it is dierentiable almost everywhere.
Therefore, even if is not specied to be dierentiable, we use |∇|
8
to denote the
Lipschitz constant of .
We summarize our assumptions as follows:
Ap,
0
q. (i) Sp,
0
q holds. (ii) For the same ,
0
,
}}|∇|
8
¤ and| detpq|¥
0
, P R
.
Note that}} is nite implies each entry|
|
0
¤}}.
Dene for P
8
0
pR
q, P R
,
pq
»
||¤1
rppqqpqp∇pqpqqs pq
||
. (2.2.1)
Proposition 2.2.1. Let Pr1, 2q, Pp0, 1q,
r
p
r
q
1¤¤
with
r
P
pR
q, |
r
|
¤
@, and Assumption Ap,
0
q hold. Let
| pq pq|¤||
for all |||| 1.
18
Then for any P
p
1
q, there exists a unique solution P
p
1
q to the parabolic
equation
B
p,q p,q
r
pq∇p,qp,q, p,qP
1
, (2.2.2)
p0,q 0, P R
.
Moreover, there is a constant p,,,,
0
q such that
||
¤||
,
and for all ¤¤ 1,
|p,qp,q|
{2 ¤pq
1{2
||
.
Proof. We apply Theorem 4 in [26] with ℒ, wherep,qP
1
,
p,q
»
rp,pqqp,qp∇p,qpqq pqs pq
||
,
p,q
pq∇p,q
»
||¡1
rp,pqqp,qs pq
||
,
and pq : 1
Pp1,2q
1
1
1
||¤1
. Set
pq
r
pq 1
Pp1,2q
pq
»
||¡1
pq
||
, P R
.
Using the symmetry assumption on and changing variables of integration, we see that
p,q
»
rp,qp,qp∇p,qq pqsp,q
||
,
where for P R
,P R
0
,
p,q
p
1
pqq
| detpq||
1
pq{|||
: r p,q p
1
pqq.
19
First we verify assumptions of Theorem 4 in [26] for p,q. Obviously,
|pq|¤||, ,P R
,
which implies||¤|pq
1
| and thus|
1
pq{|||¥ 1{, P R
, P R
0
. Therefore,
|p,q|¤
1
{
0
, P R
, P R
0
.
On the other hand, it's obvious that detpq is bounded and Lipshitz with
0
¤
|detpq| ¤
!, which implies both
1
|detpq|
and
1
pq
||
ppqq
detpq
||
are Lip-
schitz in uniformly over . With
1
¤
1
pq
||
¤
1
p 1q!
3{2
{
0
:
1
,P R,P R
0
, (2.2.3)
we can conclude r p,q is Lipschitz uniformly over . Meanwhile, recall that is -H older
continuous and 0-homogeneous. Hence for 0,,ℎP R
,
ppℎq
1
q ppq
1
q
¤
pℎq
1
|pℎq
1
|
pq
1
|pq
1
|
¤||
2 |pℎq
1
pq
1
|
|pℎq
1
|
|pq
1
|
¤ sup
∇rpq
1
s
|ℎ|
,
and therefore p,q is -continuous in uniformly over .
When 1, according to (1.2.2),
»
||¤1
p,q
||
»
||¤1
1
pq
|detpq|
1
pq
||
||
»
||¤1
1
pq
|detpq|
1
pq
||
||
0.
20
Note that, there is
2
2
p
0
,,, q such that p,q¥
2
, @P R
,@ P R
0
. Then,
Assumption A in Theorem 4 of [26] is satised.
Let t :||¡ 1u,
1
t :||¤ 1u, and p,q pq if || ¡ 1, p,q 0
otherwise. Then p,q can be written as
p,q
pq∇p,q
»
rp,p,qqp,q
p∇p,qp,qq1
1
pqs pq
||
.
By (2.2.3),||¤
1
|pq| for all ,P R
, thus|p,q|¥
1
1
for all ,P R
. Then by
choosing
1
1
, we have
»
|p,q|¤
|p,q|
pq
||
0, @P R
.
Hence, Assumption B1 of Theorem 4 in [26] holds.
We might as well set ¡ 1. Now, for|ℎ|¤ 1,
»
||¡1
r|p,qpℎ,q|^ 1s pq
||
¤
2
»
||¡1
r|ℎ|||^ 1s
||
2
»
|ℎ|||¡|ℎ|
r|ℎ|||^ 1s
||
2
|ℎ|
»
||¡|ℎ|
r||^ 1s
||
¤|ℎ|p1 1
1
|ln|ℎ||q
for some p,, q, Therefore Assumption B2 of Theorem 4 in [26] is satised and
our statement holds.
Now, consider the backward Kolmogorov equation
B
p,q
r
pq∇p,qp,q pq, p,qP
, (2.2.4)
p,q 0, P R
,
21
where is dened as (2.2.1). If solves (2.2.2) in
1
with pq, P R
, then
p,q p ,q, 1¤ ¤ ,P R
, solves (2.2.4) with Pr0, 1s. The following
statement is an obvious consequence of Proposition 2.2.1.
Corollary 2.2.2. Let Pr1, 2q, Pp0, 1q,
r
p
r
q
1¤¤
with
r
P
pR
q,|
r
|
¤, @,
and Assumption Ap,
0
q hold. Let
| pq pq|¤||
for all|||| 1.
Then for any P
pR
q and Pr0, 1s, there exists a unique solution P
p
q to
(2.2.4). Moreover, there is a constant p,,,,
0
q, independent of , such that
||
¤||
,
and for all 0¤¤¤ ,
|p,qp,q|
{2 ¤pq
1{2
||
.
2.3 Estimates of Stochastic Integrals and Driving Processes
We present here some stochastic integral estimates related to stable type point measures.
Let pFq be predictable -algebra onr0, 1q
.
Let :r0, 1s
R
0
Ñ R
be a ℬpR
0
q-measurable vector function,
pq
pq
1¤¤
, Pr0, 1s, P R
0
,
such that for any Pr0, 1s a.s.,
»
0
»
||¤1
|
pq|
2
pq
||
8. (2.3.1)
22
Let 0¤¤ ¤ 1. Consider the stochastic process
»
»
||¤1
pqp,q, Pr,s.
Note
is well dened because of (2.3.1). The following estimates hold.
Lemma 2.3.1. Let P r1, 2q, P p, 8q, 0 ¤ pq ¤ , P R
. Assume there is a
predictable nonnegative process
,Pr,s, such that
|
pq|¤
||,Pr,s, P R
.
Then there is p,,, q such that
E
sup
¤¤
|
|
¤E
»
|
|
.
Proof. If ¥ 2, then by Lemma 2.1.4(i) (e.g. Lemma 4.1 in [23]),
E
sup
¤¤
|
|
¤ E
»
»
||¤1
|
|
2
||
{2
»
»
||¤1
|
|
||
¤ E
»
|
|
. (2.3.2)
If Pp, 2q, then by the Burkholder-Davis-Gundy (BDG) inequality, see Remark 2.1.2,
E
sup
¤¤
|
|
¤ E
»
»
||¤1
2
p,q
{2
¤ E
»
»
||¤1
p,q
(2.3.3)
¤ E
»
.
23
Lemma 2.3.2. Let 0 ¤ pq ¤ , P R
. Assume there is a predictable nonnegative
process
,Pr,s, such that
|
pq|¤
||, Pr,s, P R
.
(i) Let Pp1, 2q,Pp0, q. Then there is p,,, q such that
E
sup
¤¤
|
|
¤
E
»
{ .
(ii) Let Pr1, 2q,
¤ a.s. for some constant ¡ 0 and E
³
1. Then
there is p,,, q such that
E
sup
¤¤
|
|
¤E
»
1
ln
E
»
.
Proof. For any ¡ 0,
»
»
¯
||¤,||¤1
»
»
¯
||¡,||¤1
:
1
2
, Pr,s.
Let 0 ¤ Pr1, 2q. By Remark 2.1.2 (Corollary II in [24]),
E
sup
¤¤
1
¤ E
»
»
|
¯
|¤,||¤1
|
pq|
2
||
{2
¤ E
»
»
|
¯
|¤
2
||
{2
(2.3.4)
¤
p1 {2q
E
»
{2
.
Let Pr1, 2q. Then by the BDG inequality and Remark 2.1.5,
E
sup
¤¤
2
¤ E
»
»
|
¯
|¡,||¤1
|
pq|
2
p,q
{2
24
¤ E
»
»
|
¯
|¡,||¤1
||
. (2.3.5)
If Pr1, q, Pp1, 2q, then
E
sup
¤¤
2
¤
p q
E
»
.
Taking
E
³
1{ and combining with (2.3.4) ,
E
sup
¤¤
|
|
¤
E
»
{ . (2.3.6)
If Pp0, 1q, Pp1, 2q, then by H older inequality and (2.3.6),
E
sup
¤¤
|
|
¤
E sup
¤¤
|
|
¤
E
»
{ .
If Pr1, 2q, then, according to (2.3.5),
E
sup
¤¤
2
¤ E
»
»
|
¯
|¤
||
¤ p1| ln|q E
»
.
Taking E
³
and combining with (2.3.4), we see that
E
sup
¤¤
|
|
¤E
»
1| ln
E
»
|
.
Lemma 2.3.3. Let 0 ¤ pq ¤ , P R
. Assume there is a predictable nonnegative
process
, Pr,s, such that
|
pq|¤
||, Pr,s, P R
.
25
(i) Let 1, P p0, 1q, and
¤ a.s. for some constant ¡ 0. Then there is
p,,,, q such that
E
sup
¤¤
|
|
¤
E
»
1
ln
E
»
.
(ii) Let 1, P p0, 1q. Assume pq pq, P R
. Suppose there exists a
predictable matrix valued function
,Pr,s, such that a.s.
|
pq
|¤
||
1 1
, Pr,s, ||¤ 1,
for some constants ¡ 0, 1
¡ 0. Then there is p,,,, q such that
E
sup
¤¤
|
|
¤
E
»
.
Proof. (i) Let 1, Pp0, 1q. By the H older inequality,
E
sup
¤¤
|
|
¤
E sup
¤¤
|
|
,
and the estimate follows by Lemma 2.3.2(ii).
(ii) For ¡ 0, we decompose
»
»
¯
||¤,||¤1
»
»
¯
||¡,||¤1
:
1
2
, Pr,s.
Let 0 1. By Remark 2.1.2 (Corollary II in [24]), there is p,,q such that
E
sup
¤¤
1
¤ E
»
»
|
¯
|¤
2
||
1
{2
(2.3.7)
¤
{2
E
»
{2
.
26
We decompose further
2
»
»
¯
||¡,||¤1
pqp,q
»
»
¯
||¡,||¤1
p
pqq pq
||
1
:
21
22
, Pr,s.
Now,
E
sup
¤¤
21
¤ E
»
»
¯
||¡,||¤1
p,q
¤ E
»
»
¯
||¡
||
1
¤
p1q
E
»
,
and
E
sup
¤¤
22
¤ E
»
»
¯
||¡,||¤1
|
pq
|
||
1
¤ E
»
»
||¤1
||
1 1
||
1
¤ E
»
¤
E
»
.
Combining these estimates with (2.3.7) and taking E
³
, we see that for
1, Pp0, 1q, there is p,,,, q such that
E
sup
¤¤
|
|
¤
E
»
.
Again, let :r0, 1s
R
0
Ñ R
be a ℬpR
q-measurable vector function,
pq
pq
1¤¤
, Pr0, 1s, P R
0
,
27
such that for any Pr0, 1s a.s.,
»
0
»
||¡1
|
pq| pq
||
8 if Pr1, 2q. (2.3.8)
Let 0¤¤ ¤ 1. Consider the stochastic process
»
»
||¡1
pqp,q, Pr,s.
Note
is well dened because of (2.3.8).
Later we will need the following estimates as well.
Lemma 2.3.4. Let Pr1, 2q, Pp0, q, 0¤ pq¤ , P R
, 0¤ ¤ ¤ 1. Assume
there is a predictable nonnegative process
, Pr,s, such that
|
pq|¤
||,Pr,s,P R
.
Then there is p,,, q such that
E
sup
¤¤
|
|
¤E
»
.
Proof. Let Pp0, 1q. Then, according to Remark 2.1.5,
E
sup
¤¤
|
|
¤ E
»
»
||¡1
|
pq|
pq
||
¤E
»
.
Let Pr1, q, Pp1, 2q. By Lemma 2.1.4(ii),
E
sup
|
|
¤ E
»
»
||¡1
||
»
»
||¡1
||
¤ E
»
»
¤ E
»
.
28
We now apply Lemmas 2.3.1-2.3.3 to estimate
0
»
0
»
||¤1
p,q,Pr0, 1s.
Lemma 2.3.5. Let 0¤ pq¤.
(i) There is p,,, q such that for all Pr0, 1s,
E
0
¤ if ¡ Pr1, 2q,
E
0
¤ p1|ln|q if Pr1, 2q,
E
0
¤
{ if Pp1, 2q,
and
E
0
¤
p1|ln|q
, 1.
(ii) Let 1 and pq pq, P R
. There is p,,q such that for all
Pr0, 1s,
E
0
¤
if 1.
Proof. These estimates are obvious consequences of Lemmas 2.3.1 - 2.3.3 when they are
applied to
pq,P R
.
Now we estimate
0
»
0
»
||¡1
p,q 1
Pp1,2q
»
||¡1
pq
||
, Pr,s.
Lemma 2.3.6. Let 0¤ pq¤.
(i) For each Pp0, q there is p,,, q such that for all Pr0, 1s,
Er|
|
s¤
{ if Pp1, 2q,
29
and
Er|
|
s¤
p1|ln|q
if 1.
(ii) If 1, pq pq, P R
. Then for each Pp0, q there is p,,q
such that
Er|
|
s¤
, Pr0, 1s.
(iii) If Pr1, 2q, then there is p,, q such that
Er|
|
^ 1s¤p1| ln|q, Pr0, 1s.
Proof. The estimates in (i)-(ii) are obvious consequences of Lemmas 2.3.5 and 2.3.4 applied
to
pq, P R
. We prove (iii) only.
Let
»
0
»
||¡1
p,q,
1
Pp1,2q
»
0
»
||¡1
pq
||
,
i.e.,
0
, Pr0, 1s. According to Lemma 2.3.5, there is p,, q so that
E
0
¤p1| ln|q, Pr0, 1s.
Now,
|
|
^ 1
»
0
»
||¡1
rp|
|
^ 1qp|
|
^ 1qsp,q
¤
»
0
»
||¡1
p||^ 1qp,q, Pr0, 1s.
Hence
Er|
|
^ 1s¤, Pr0, 1s.
30
Obviously,|
|¤,Pr0, 1s. Hence (iii) holds.
A straightforward consequence of Lemma 5.1.10 is the following statement.
Corollary 2.3.7. Let Pr1, 2q, 0¤ pq¤,||¤,}}¤.
(i) For each Pp0, q, there is p,,, q such that for all Pr0, 1s,
E
pq
¤
{ if Pp1, 2q,
and
E
pq
¤p{ lnq
if 1.
(ii) If 1, pq pq, P R
. Then for each Pp0, q there is p,,q
such that for all Pr0, 1s,
E
pq
¤
.
(iii) There is p,, q such that for all Pr0, 1s,
E
pq
^ 1
¤p{ lnq
1
.
Proof. It is obviously true when 0. For@Pp0, 1s, there is Pt0, 1,..., 1u so that
{ ¤ p 1q{, and
pq : {. Thus 0
pq ¤ 1{. Note that for any
,¡ 0,
in distribution. All the estimates immediately follow from Lemma
5.1.10.
Finally, applying Lemma 2.3.5 we derive
Corollary 2.3.8. Let 0¤ pq¤.
(i) There is p,,, q such that for all Pr0, 1s,
E
pq
¤
$
'
'
'
'
&
'
'
'
'
%
1
if ¡ Pr1, 2q,
p{ lnq
1
if Pr1, 2q,
{ if Pp1, 2q,
31
and
E
pq
¤p{ lnq
if 1.
(ii) Let 1, pq pq, P R
. There isp,,q such that for allPr0, 1s,
E
pq
¤
if 1.
Proof. It is obviously true when 0. For@Pp0, 1s, there is Pt0, 1,..., 1u so that
{ ¤ p 1q{, and
pq : {. Thus 0
pq ¤ 1{. Note that for any
,¡ 0,
0
0
0
in distribution. All the estimates immediately follow from Lemma
2.3.5.
32
Chapter 3
Proof of Main Results
We start with the Lipschitz and possibly completely degenerate case and derive the rate of
convergence directly.
3.1 Lipschitz Drift and Non-truncated Noise
Note that
0
1
Pp1,2q
»
||¡1
pq
||
, Pr0, 1s,
where
»
0
»
||¡1
p,q, Pr0, 1s.
Denote
r
pqpq 1
Pp1,2q
pq
»
||¡1
pq
||
, P R
.
Let
be the strong solution to (1.2.1) and
:
,Pr0, 1s. Let 0¤¤ ¤ 1.
Then
»
r
r
pq
r
p
qs
»
r
r
p
q
r
p
qs
»
r
pq
s
0
»
r
s
0
»
r
pq
s
»
r
s
:
6
¸
1
, Pr,s.
33
Estimates of
1
and
2
. For Pr1, q, Pp1, 2q, by the H older inequality,
E
sup
¤¤
1
2
¤ E
»
r
pq
^ 1s
»
¤ E
»
r
pq
^ 1spq
sup
¤¤
,
and for Pp0, 1q, Pr1, 2q,
E
sup
¤¤
|
1
2
|
¤
E
»
r|
pq
|^ 1s
pq
E
sup
¤¤
|
|
for some p,,, q. By Corollary 2.3.7, for P p0, q there is p,,, q
such that
E
sup
¤¤
1
2
¤pq
{ pq
E
sup
¤¤
|
|
,
where pq if Pp0, q, Pp1, 2q, and pq{ ln if 0 1.
Estimate of
3
. By denition,
3
»
»
||¤1
pq
p,q,Pr,s.
According to Corollary 2.3.7, there is p,, q so that
: E
»
r
pq
^ 1s¤p{ lnq
1
.
Apply Lemma 2.3.2 with
pq
, P r0, 1s, then for all P
p0, q, Pp1, 2q, there is p,, q such that
E
sup
¤¤
3
¤
{ p{ lnq
{ .
34
If 1, Pp0, 1q, by Lemma 2.3.3(i),
E
sup
¤¤
3
¤
p1|ln|q
¤p{ lnq
r1 lnp{ lnqs
¤
{plnq
2
.
If 1, P p0, 1q, and pq pq, P R
, then applying Lemma 2.3.3(ii) with
pq
, we have
E
sup
¤¤
3
¤p{ lnq
.
Estimate of
4
. By denition,
4
»
»
||¤1
p
q
p,q, Pr,s.
According to Remark 2.1.2, for Pp0, 2q, there is p,q so that
E
sup
¤¤
4
¤E
»
2
{2
¤pq
{2
E
sup
¤¤
.
Estimate of
5
. By denition,
5
»
»
||¡1
r
pq
sp,q, Pr,s.
Applying Lemma 2.3.4 with
pqrp
pq
qp
qs,
|p
pq
qp
q|,
Pr0, 1s, P R
, and combining
2
pq
^ 1
, Pr0, 1s,
we can conclude for Pp0, q that
E
sup
¤¤
5
¤E
»
p
pq
^ 1q.
35
Hence by Corollary 2.3.7,
E
sup
¤¤
5
¤ p{ lnq
if 1,
E
sup
¤¤
5
¤
{ if 0 Pp1, 2q.
Estimate of
6
. By denition,
6
»
»
||¡1
p
q
p,q, Pr,s.
By Lemma 2.1.4 (ii), for Pr1, q, Pp1, 2q there is p,,, q so that
E
sup
¤¤
6
¤ E
»
»
¤ E
»
¤pq E
sup
¤¤
.
According to Remark 2.1.5, for Pp0, 1q there is p,,, q such that
E
sup
¤¤
6
¤pq E
sup
¤¤
.
Summarizing, for Pp0, q there is p,,, q so that for any ¤¤ ¤ 1,
E
sup
¤¤
¤
"
E
p{ lnq
{ pq
{2
E
sup
¤¤
*
if Pp1, 2q, and
E
sup
¤¤
¤
"
E
{plnq
2
pq
{2
E
sup
¤¤
*
if 1. If 1, and pq pq,P R
, then
E
sup
¤¤
¤
"
E
p{ lnq
pq
{2
E
sup
¤¤
*
.
36
If pq
{2
¤ 1{2, then there is
r
r
p,,, q such that for Pp0, q,
E
sup
¤¤
¤
r
tE
p{ lnq
{ u if Pp1, 2q,
E
sup
¤¤
¤
r
tE
{plnq
2
u if 1,
E
sup
¤¤
¤
r
tE
p{ lnq
u if 1 with symmetry .
The claim now follows by Lemma 2.1.6.
3.2 Lipschitz Drift and Truncated Noise
Let
be the strong solution to (1.2.4) and
, Pr0, 1s. Let 0¤ ¤ ¤ 1.
Then
»
r
pq
p
qs
»
rp
qp
qs
»
r
pq
s
0
»
r
p
qs
0
:
1
2
3
4
, Pr,s.
Estimates for P p0, q of
, 1,..., 4, are identical to estimates of
,
1,..., 4, and the conclusion of section 3.1 holds for Pp0, q with
replaced by
.
Estimates of
1
and
2
for Pr, 8q. By H older inequality and Corollary 2.3.8,
Er sup
¤¤
1
s ¤ E
»
pq
¤p{ lnq
1
,
Er sup
¤¤
1
s ¤ E
»
pq
¤
1
if ¡.
By H older inequality, for Pr, 8q there is pq such that
Er sup
¤¤
2
s¤E
»
¤pq Er sup
¤¤
s.
37
Estimate of
3
for Pr, 8q. By denition,
3
»
»
||¤1
pq
p,q, Pr,s.
By Corollary 2.3.8, there is p,, q so that
: E
»
pq
¤p{ lnq
1
.
Applying Lemma 2.3.2(ii) with
pq
, Pr0, 1s, we can claim there
is p,, q such that
E
sup
¤¤
3
¤ p1 lnq¤p{ lnq
1
r1 lnp{ lnqs
¤
{plnq
2
1
.
Now, for ¡, by Lemma 2.3.1 and Corollary 2.3.8, there is p,,, q such that
E
sup
¤¤
3
¤E
»
¤
1
.
Estimate of
4
for Pr, 8q. By denition,
4
»
»
||¤1
p
q
p,q, Pr,s.
By Lemma 2.1.4(i) (Kunita's inequality) and Remark 2.1.2, there is p,,, q such
that
E
sup
¤¤
4
¤ E
»
2
{2
»
¤
pqpq
{2
E
sup
¤¤
.
38
Summarizing, there is p,,, q so that for any ¤¤ ¤ 1,
E
sup
¤¤
¤
"
E
{plnq
2
1
pq
{2
E
sup
¤¤
*
,
and for all ¡ ,
E
sup
¤¤
¤
"
E
1
pq
{2
E
sup
¤¤
*
.
We nish the proof by taking pq
{2
¤ 1{2 and applying Lemma 2.1.6.
3.3 H¨ older Drift and Non-truncated Noise
First we prove that the Euler approximation sequence is a Cauchy sequence.
Lemma 3.3.1. Let P r1, 2q, P p0, 1q, ¡ 1 {2, P p0, q and Ap,
0
q hold.
Assume, without loss of generality, | |
¤ , ||
¤ for the same . Then there are
constants
1
1
p,,,,
0
,q,
1
1
p,,,,
0
,q such that for any 0¤¤ ¤ 1
with ¤
1
we have
E
sup
¤¤
|
|
¤
1
Er|
|
s
{
{
.
Moreover, if
is a strong solution to (1.2.1), then
E
sup
¤¤
|
|
¤
1
Er|
|
s
{
.
Proof. By Corollary 2.2.2, for each 1,...,, there exists a unique solution
p,q to
(2.2.4) with
r
pqpq 1
Pp1,2q
pq
»
||¡1
pq
||
, P R
.
39
Note that
r
is also a bounded -H older continuous function. Denote p
q
1¤¤
. By
the It^ o formula and denition of Euler approximation (1.2.3), for Pr,s, using (2.2.4)
for
r
,
p,
q
p,
q
»
r
p
q
»
r
pq
r
p
q
∇
p,
q
»
»
||¤1
,
pq
,
p,q
»
»
||¡1
,
pq
,
p,q
»
»
||¤1
t
,
pq
p,
p
qq
∇
p,
q
pq
p
q
u pq
||
.
On the other hand, according to (1.2.3), for Pr,s
»
r
p
q
»
r
pq
r
p
q
»
»
||¡1
pq
p,q
»
»
||¤1
pq
p,q.
It follows from the two identities above that
7
¸
1
,
,
where
,1
p,
qp,
q
,
,2
»
∇p,
q
r
pq
r
p
q
,
,3
»
»
||¤1
tp,
p
qq
,
pq
40
∇p,
q
p
q
pq
u pq
||
,
,4
»
»
||¤1
!
r
,
,
pq
s
r
pq
s
)
p,q,
,5
»
»
||¡1
r
,
,
pq
s
r
pq
s
)
p,q,
,6
»
»
||¤1
!
,
,
)
p,q,
,7
»
»
||¡1
!
,
,
)
p,q.
Let
,;
,
,
and
,
, ,¥ 1, 1,..., 7.
Estimate of
,;1
. Using the terminal condition of (2.2.4) and Corollary 2.2.2, we see
that for ¡ 0 there is a constant p,,,,,
0
q such that
,;1
¤ t|
|
|p,
qp,
qp,
qp,
q|
|p,
qp,
qp,
qp,
q|
u
¤ tr|
|
spq
{2
|
|
u,
therefore,
E
sup
¤¤
,;1
¤tpq
{2
E
sup
¤¤
|
,
|
E
,
u.
Estimate of
,;2
. Obviously,
,;2
¤ 2
r
,2
,2
s, Pr0, 1s.
For P r1, q, P p1, 2q, by H older inequality and Corollary 2.3.7, there is
p,,,
0
,,q such that
E
sup
¤¤
,2
¤E
»
pq
¤
{ .
41
For Pp0, 1q, by Corollary 2.3.7, there is a constant p,,,, q such that
E
sup
¤¤
,2
¤ E
»
pq
¤
»
Er
pq
s
¤
{ .
Similarly, we can obtain the estimates for
,2
. Hence, by H older inequality, for all
Pp0, q, there is p,,,, q such that
E
sup
¤¤
,;2
¤r
{
{ s.
Estimate of
,;3
. Obviously,
,;3
¤ 2
r
,3
,3
s,Pr0, 1s. Note that
,3
»
»
||¤1
!
r
»
1
0
r∇p,
q∇
,
p
q
pq
p
q
s
pq
p
q
)
pq
||
.
Let 1
Pp0, 1q, ¡ 1 1
¡ and denote
pq
, Pr0, 1s.
Then there is p,,,
0
, q such that
,3
¤
»
»
||¤1
||
1 1
||
¤
»
pq
^ 1, Pr,s.
Hence by Corollary 2.3.7, for Pr1, q, Pp1, 2q,
E
sup
¤¤
,3
¤
»
E
pq
¤
{ ;
for Pp0, 1q, according to Corollary 2.3.7,
E
sup
¤¤
,3
¤
»
E
pq
^ 1
¤p{ lnq
{ .
42
Similar reasoning can be applied to
,3
. Therefore for all P p0, q there is
p,,,, q such that
E
sup
¤¤
,;3
¤
p{ lnq
{ p{ lnq
{
.
Estimate of
,;4
. Obviously,
,;4
¤ 2
r
,4
,4
s,Pr0, 1s. By Corollary
2.3.7(iii), there is p,, q such that
: E
»
pq
^ 1
¤p{ lnq
1
.
First, let Pp1, 2q. Applying Lemma 2.3.2(i) to
,4
with
2p1|∇|
0
q
pq
^ 1
,Pr,s, (3.3.1)
and Corollary 2.3.7(iii), we have that for P p0, q there is p,,,,,
0
q such
that
E
sup
¤¤
,4
¤
{ ¤p{ lnq
{ .
Now, let 1. Applying Lemma 2.3.3(i) to
,4
with
given by (3.3.1), and Corollary
2.3.7(iii), we see there is p,,,, q such that
E
sup
¤¤
,4
¤
p1|ln|q
¤
{plnq
2
.
Similarly,
E
sup
¤¤
,4
¤
{plnq
2
{ ,
and thus there is p,,,,,
0
q so that
E
sup
¤¤
,;4
¤
"
{plnq
2
{
{plnq
2
{ *
.
43
Estimate of
,;5
. Obviously,
,;5
¤ 2
r
,5
,5
s, Pr0, 1s. By Lemma
2.3.4, applied to
,5
with
p1|∇|
0
q|∇|
8
pq
, Pr,s,
and Corollary 2.3.7, there is p,,,,,
0
q such that
E
sup
¤¤
,5
¤
{ for Pp0, q.
Similarly as above, for Pp0, q,
E
sup
¤¤
,;5
¤
{
{
.
Estimate of
,;6
. Denote
,
p
qp
q, Pr,s. Then
,;6
»
»
||¤1
!
r
s
,
,
)
p,q
»
»
||¤1
tr
,
,
s
rp,
qp,
qsup,q
:
,;61
,;62
.
For Pp0, 2q, by Remark 2.1.2, there is p,,,,
0
,q such that
E
sup
¤¤
,;61
¤ E
»
»
||¤1
|
,
|
2
||
{2
¤ pq
{2
E
sup
¤¤
|
,
|
.
44
We rewrite
,;62
»
»
||¤1
»
1
0
r∇
,
,
∇
,
,
s
,
p,q,Pr,s.
Let 1 1
and 2 1
¡ . Then by Remark 2.1.2, there is p,,,,
0
,q
such that
E
sup
¤¤
,;62
¤ E
»
|
,
|
2
»
||¤1
||
2 1
||
{2
¤ pq
{2
E
sup
¤¤
|
,
|
.
Hence,
E
sup
¤¤
,;6
¤pq
{2
E
sup
¤¤
|
,
|
.
Estimate of
,;7
. Let P p1, 2q, P r1, q. By Lemma 2.1.4(ii) (see Lemma 4.1
in [23]), there is p,,,
0
,,q such that
E
sup
¤¤
,;7
¤ E
»
»
||¡1
r|
,
||
,
|s
||
»
»
||¡1
r|
,
|
|
,
|
s
||
¤ pq E
sup
¤¤
|
,
|
.
Let 1,Pp0, 1q. By Remark 2.1.5, there is p,,,, q such that
,;7
¤
»
»
||¡1
r
,
,
s
p,q,Pr,s,
and thus
E
sup
¤¤
,;7
¤ E
»
»
||¡1
r|
,
|
|
,
|
s
||
45
¤ pq E
sup
¤¤
|
,
|
.
Collecting all the estimates above we see that for P p0, q there is
p,,
0
,,,q such that
E
sup
¤¤
|
,
|
¤
!
pq
{2
E
sup
¤¤
|
,
|
E
,
{
{ )
. (3.3.2)
Set
1
p2q
2{
,
1
2 with the in (3.3.2), we then have
E
sup
¤¤
|
,
|
¤
1
!
E
,
{
{ )
if 0¤¤
1
.
Rate of convergence. Now let us assume
is a strong solution to (1.2.1). We have, by
It^ o formula and (2.2.4), for Pr,s,
p,
qp,
q
»
r
p
q
»
»
||¤1
rp,
p
qqp,
qsp,q
»
»
||¡1
rp,
p
qqp,
qsp,q,
Hence for Pr,s, we obtain
p,
qp,
q
»
»
||¤1
tp
qrp,
p
qqp,
qsup,q
»
»
||¡1
tp
qrp,
p
qqp,
qsup,q,
46
and
t
rp,
qp,
qsrp,
qp,
qsu
5
¸
2
,
,6
,7
»
»
||¤1
tp
qrp,
p
qqp,
qsup,q
»
»
||¡1
tp
qrp,
p
qqp,
qsup,q.
Estimates for
,
, 2,..., 5 have been derived above. And we can estimate
rp,
qp,
qsp,
qp,
q,
,6
»
»
||¤1
tp
qrp,
p
qqp,
qsup,q,
,7
»
»
||¡1
tp
qrp,
p
qqp,
qsup,q
in exactly the same way as we estimated
,;1
,
,;6
and
,;7
(by replacing
by
in the arguments). We nd that there is a constant p,,,,
0
,q such that
E
sup
¤¤
|
|
¤
1
Er|
|
s
{
.
Then the claimed rate of convergence holds because of Lemma 2.1.6.
Existence of a solution. Let P p0, q and
1
be the constant in Lemma 3.3.1. By
Lemmas 2.1.6 and 3.3.1, there is p,,,,
0
,q such that for ,¥ 1,
E
sup
0¤¤1
|
|
¤
{
{
,
and thus
E
sup
0¤¤1
|
|
Ñ 0
47
as ,Ñ8. Therefore there is an adapted c adl ag process
such that for all Pp0, q,
E
sup
0¤¤1
|
|
Ñ 0
asÑ8. Hence
solves (1.2.1). Moreover, by Lemma 3.3.1, there isp,,,, q
such that
E
sup
0¤¤1
|
|
¤
{ .
Uniqueness follows from Lemma 3.3.1: any strong solution can be approximated by
.
3.4 H¨ older Drift and Truncated Noise
The proof repeats the steps we took to prove Proposition 1.2.1.
Lemma 3.4.1. Let Pr1, 2q, Pp0, 1q, ¡ 1 {2, ¡ 0 and Ap,
0
q hold. Assume
(without loss of generality),| |
¤ ,||
¤ for the same . Then there are constants
1
1
p,,,,
0
,q,
1
1
p,,,,
0
,q such that for any 0¤ ¤ ¤ 1 with
¤
1
we have
E
sup
¤¤
|
|
¤
1
pEr|
|
sp,,, qp,,, qq,
wherep,,, q
{ if ,p,,, qp{ lnq
1
if , andp,,, q
1
if ¡.
Moreover, if
is a strong solution to (1.2.3), then
E
sup
¤¤
|
|
¤
1
tEr|
|
sp,,, qu.
Proof. Let 0¤ ¤ ¤ 1. By Corollary 2.2.2, for each 1,...,, there exists a unique
solution
p,q to (2.2.4) with
r
pq pq pq, P R
. Denote
1¤¤
.
48
By the It^ o formula and denition of the Euler approximation (1.2.5), for Pr,s, using
(2.2.4),
p,
q
p,
q
»
p
q
»
pq
p
q
∇
p,
q
»
»
||¤1
,
pq
,
p,q
»
»
||¤1
t
,
pq
p,
p
qq
∇
p,
q
pq
p
q
u pq
||
.
On the other hand, according to (1.2.5), for Pr,s
»
p
q
»
pq
p
q
»
»
||¤1
pq
p,q.
It follows from the two identities above that
4
¸
1
,
,5
,
where
,1
p,
qp,
q
,
,2
»
∇p,
q
pq
p
q
,
,3
»
»
||¤1
tp,
p
qq
,
pq
∇p,
q
p
q
pq
u pq
||
,
,4
»
»
||¤1
!
r
,
,
pq
s
r
pq
s
)
p,q,
,5
»
»
||¤1
!
,
,
)
p,q.
49
Let
,;
,
,
, and
,
,,¥ 1, 1,..., 5.
Estimate of
,;1
. This estimate is identical to that of
,;1
in the proof of Lemma
3.3.1. Repeating it and applying Corollary 2.2.2, we see that for Pp0,8q there is
p,,,,
0
,q so that
,;1
¤tr|
|
spq
{2
|
|
u,
and,
E
sup
¤¤
,;1
¤tpq
{2
E
sup
¤¤
|
,
|
E
,
u.
Estimates of
,;
, 2, 3, 4, forPp0, q are identical to the estimates of
,;
,
2, 3, 4. We replace by , and apply Corollary 2.3.8 instead of 2.3.7. Note that for
Pp0, q the estimates in Corollary 2.3.7 coincide with estimates in Corollary 2.3.8. Hence
for Pp0, q there is p,,,
0
,q such that for 2, 3, 4,
E
sup
¤¤
,;
¤
{
{
.
Estimate of
,;2
for Pr, 8q. By H older inequality and Corollary 2.2.2, there is
p,,,,
0
,q such that
E
sup
¤¤
,2
¤E
»
pq
.
Hence, by Corollary 2.3.8,
E
sup
¤¤
,2
¤p,,, q,
wherep,,, q
{ if ,p,,, qp{ lnq
1
if , andp,,, q
1
if ¡. Therefore for Pr, 8q,
E
sup
¤¤
,;2
¤rp,,, qp,,, qs.
50
Estimate of
,;3
for Pr, 8q. By repeating the argument for
,3
in the proof of
Proposition 1.2.1, we nd that there is p,,, q so that
,3
¤
»
pq
,Pr,s.
Hence by Corollary 2.3.8, for ¥ there is p,,,,
0
,q such that
E
sup
¤¤
,3
¤
»
E
pq
¤p, 1,, q.
Therefore, for Pr, 8q there is p,,,,
0
,q such that
E
sup
¤¤
,;3
¤ rp, 1,, qp, 1,, qs
¤ rp,,, qp,,, qs.
Estimate of
,;4
for Pr, 8q.
,4
»
»
||¤1
!
r
,
,
pq
s
r
pq
s
)
p,q,
By Corollary 2.3.8(i), there is p,, q such that
: E
»
pq
¤p{ lnq
1
.
Applying Lemma 2.3.2(ii) with
p1|∇|
0
q|∇|
8
pq
,Pr,s, (3.4.1)
51
we can see there is p,,,
0
,q such that
E
sup
¤¤
,4
¤p1|ln|q¤
{plnq
2
1
.
By Lemma 2.3.1 with
given by (3.4.1) and Corollary 2.3.8, for ¡ there is
p,,,,,
0
q such that
E
sup
¤¤
,4
¤
1
.
Hence for ¥ there is p,,,,,
0
q such that
E
sup
¤¤
,;4
¤rp,,, qp,,, qs.
Estimate of
,;5
. As in the case of
,;6
in the proof of Proposition 1.2.1, we rewrite
,;5
»
»
||¤1
!
r
s
,
,
)
p,q
»
»
||¤1
tr
,
,
s
rp,
qp,
qsup,q
:
,;51
,;52
,
and
,;52
»
»
||¤1
»
1
0
r∇
,
,
∇
,
,
s
,
p,q,Pr,s.
52
For Pp0, 2q, repeating the estimates of
,;6
in the proof of Proposition 1.2.1, we
nd that for Pp0, 2q there is p,,,
0
,, q such that
E
sup
¤¤
,;5
¤pq
{2
E
sup
¤¤
|
,
|
.
For ¥ 2, by Lemma 2.1.4(i), there is p,,,
0
,, q such that
E
sup
¤¤
,;5
¤pq E
sup
¤¤
|
,
|
.
Collecting all the estimates above we see that for P p0,8q there is
p,,,
0
,,q such that
E
sup
¤¤
|
,
|
¤
!
rpq
{2
pqsE
sup
¤¤
|
,
|
E
,
p,,, qp,,, q
)
.
There is
1
1
p,,,
0
,,q such that
pq
{2
pq
¤ 1{2 if 0 ¤
¤
1
. In that case
E
sup
¤¤
|
,
|
¤ 2
!
E
,
p,,, qp,,, q
)
.
Rate of convergence. Now let us assume
is a strong solution to (1.2.4). We have, by
It^ o formula and (2.2.4), for Pr,s,
p,
qp,
q
»
p
q
»
»
||¤1
rp,
p
qqp,
qsp,q.
Hence for Pr,s, we obtain
p,
qp,
q
53
»
»
||¤1
tp
qrp,
p
qqp,
qsup,q,
and thus
t
rp,
qp,
qsrp,
qp,
qsu
4
¸
2
,
,5
»
»
||¤1
tp
qrp,
p
qqp,
qsup,q.
Estimates for
,
, 2, 3, 4 have been derived above. And we can estimate
rp,
qp,
qsp,
qp,
q,
,6
»
»
||¤1
tp
qrp,
p
qqp,
qsup,q
in exactly the same way as we estimated
,;1
and
,;5
(by replacing
by
in the
arguments). We nd that there is a constant p,,,,
0
,q such that
E
sup
¤¤
|
|
¤rEr|
|
sp,,, qs,
and the claimed rate of convergence holds by Lemma 2.1.6.
The existence and uniqueness part is a simple repeat of the arguments in the proof of
Proposition 1.2.1.
54
Part II
The Cauchy Problem for Parabolic
Non-degenerate Kolmogorov Equations in
Generalized H¨ older Spaces
55
Chapter 4
Introduction
4.1 Motivation and Problem Description
We have seen an application of Kolmogorov integro-dierential equation in Part I, where the
solution to a Kolmogorov integro-dierential equation was used to replace the less regular
drift coecient in a SDE. To see another application, let us recall the option pricing models
with jumps in the area of mathematical nance.
The value of a European option
with a terminal payo p
q is dened as a dis-
counted conditional expectation of its terminal payo under risk-neutral probability [10],
E
p q
p
q|ℱ
E
p q
p
q|
(Markov Property)
: p, q. (4.1.1)
where
:
0
p q
, is a L evy process under the risk-neutral measure such that
is a martingale and ℱ
is the augmented ltration generated by t
: 0 ¤ ¤ u. Set
, lnp{
0
q, pqp
0
q{
0
and p,q
,
0
{
0
,
thenp4.1.1q can be written as
p,q Erp
qs,
which, according to Denition 1.1.7, is a solution of
B
p,qp,q,
56
and is the innitesimal generator of
. Clearly, solving such equations will help under-
stand the distributional behavior of
and thus
.
LetA
be the collection of L evy measures on R
with order , namely, if PA
,
inf
#
2 :
»
||¤1
||
8
+
, Pp0, 2q.
We assume that for PA
,
»
||¡1
|| 8 if Pp1, 2q,
»
||¤
1
0 for all 0
1
8 if 1.
In this part we consider the parabolic Cauchy problem with ¥ 0
B
p,q p,q p,qp,q, (4.1.2)
p0,q 0, p,qP
:r0,s R
,
and integro-dierential operator
pq
pq
»
r pq pq pq∇ pqs pq, P
8
0
pR
q,
where PA
, pq : 1
Pp1,2q
1
1
1
||¤1
. Given such , there exist a Poisson random
measure p,q onr0,8q R
such that
Erp,qs pq,
and a L evy process
so that
»
0
»
R
0
pq
r
p,q
»
0
»
R
0
p1 pqqp,q, ¥ 0, (4.1.3)
57
with
r
p,qp,q pq. For PA
, set
pq pq pt||¡uq,¡ 0,
pq
pq pq
1
,¡ 0.
Our main assumption is that pq
pq pq
1
, ¡ 0, is an O-RV function (O-
regular variation function) at zero (see [2] and [7]), that is
1
pq lim
Ñ0
pq
1
pq
1
8, ¡ 0.
By Theorem 2 in [2], the following limits exist:
1
1
lim
Ñ0
log
1
pq
log
,
1
1
lim
Ñ8
log
1
pq
log
, (4.1.4)
and
1
¤
1
. It can be shown (see Remark 4.2.5) that
1
¤ ¤
1
. In this paper, we
study the Cauchy problem (4.1.2) in the scale of spaces of generalized H older functions
whose regularity is determined by the L evy measure . We use to dene generalized
Besov norms ||
, 8
and generalized spaces
r
8,8
p
q, ¡ 0 (See Section 2.2.). They
are Besov spaces of generalized smoothness (see [19], [20], [15]) with admissible sequence
,¥ 0, and covering sequence
, ¥ 0, ¡ 1. In particular (see Section 2),
for Pp0,
1
1
q, the norm||
, 8
for the functions on R
is equivalent to
||||
sup
|pq| sup
|pqpq|
p||q
.
When is \close"to an -stable measure, they reduce to the classical Besov (or equiv.
H older-Zygmund) spaces.
Let
r
pqpq pq, Pr0, 1s.
58
The main result of Part II is
Theorem 4.1.1. Let ¡ 0, ¥ 0. Let PA
, and
be an O-RV function at zero
with
1
,
1
defined in (4.1.4). Assume
A.
0
1
¤
1
1 if Pp0, 1q, 1¤
1
¤
1
2 if 1,
1
1
¤
1
2 if Pp1, 2q ;
B.
inf
Pp0,1s,|
p
|1
»
||¤1
p
2
r
pq¡ 0;
C. There is
0
¡ 2 so that
»
8
1
pq
1
1
0
8.
Then for each P
r
8,8
p
q there is a unique solution P
r
1 8,8
p
q solving (4.1.2).
Moreover,
||
, 8
¤ pq||
, 8
, (4.1.5)
||
1, 8
¤ r1 pqs ||
, 8
(4.1.6)
and
p,q
1
,
, 8
¤
!
1
1 r1 pqs
1
)
||
, 8
for any Pr0, 1s and
1
¤, where pq
1
^
. The constant does not depend
on ,,, .
More specic examples could be the following.
59
Example 1. According to [12], Chapter 3, 70-74, any L evy measure PA
can be disin-
tegrated as
pq
»
8
0
»
1
Γ
pq p,q pq, Pℬ
R
0
,
where , and p,q,¡ 0, is a measurable family of measures on the unit sphere
1
with p,
1
q 1,¡ 0. If is an O-RV function,|tPr0, 1s :
1
pq 1u|¡ 0, A,
C and
inf
|
p
|1
»
1
p
2
p,q¥
0
¡ 0, ¡ 0,
hold, then all assumptions of Theorem 4.1.1 are satised (see Corollary 4.2.4).
Example 2. Consider L evy measures in radial and angular coordinate system (||,
||
) in the form
pq
»
8
0
»
||1
1
pqp,qpq
1
pq,Pℬ
R
0
,
where pq is a nite measure on the unit sphere.
Assume
(i) There is ¡ 1,¡ 0, 0 1
¤ 2
1, such that
1
2
¤pq
¤
2
and for all 1 ¤,
1
1
¤
pq
pq
¤
2
.
(ii) There is a function 0
pq dened on the unit sphere such that 0
pq¤ p,q¤
1,@¡ 0, and for all
p
1,
»
1
p
2
0
pqpq¥¡ 0.
60
Under these assumptions, it can be shown that B and C hold, and
is an O-RV function
with 2 1
¤
1
¤
1
¤ 2 2
. Among the options for could be (see [22])
(1) pq
1
,
Pp0, 1q, 1,...,;
(2) pqp
q
, , Pp0, 1q;
(3) pq
plnp1qq
, Pp0, 1q, Pp0, 1 q;
(4) pqp
1{ q
, Pp0, 1q, ¡ 0;
(5) pqrlnpcosh
?
qs
, Pp0, 1q.
Equations in classical H older spaces with non-local nondegenerate operators of the form
ℒpq 1
Pp0,2q
»
pqpq 1
¥1
1
||¤1
∇pq
p,q pq
1
2
pqB
2
pq 1
¥1
r
pqB
pqpqpq, P R
,
were considered in many papers. In [1], the existence and uniqueness of a solution to a
parabolic equation withℒ in H older spaces was proved analytically for H older continuous
in and smooth in , pq {||
. The elliptic problem ℒ in R
with
pq {||
was considered in [5], [8] and [13]. In [8], the interior H older estimates
(in a non-linear case as well) were studied assuming that is symmetric in . In [5], with
pq {||
, the a priori estimates were derived in H older classes assuming H older
continuity of in , except the case 1. Similar results, including the case 1 were
proved in [13]. In [9] (see references therein), in the classical H older spaces the case of a
nondegenerate
pq
»
8
0
»
1
Γ
pqp,qpq
1 , PℬpR
0
q,
with a nite measure pq on the unit sphere was considered. Finally, in [35], for (4.1.2)
with -dependent density p,q at , under dierent assumptions than A-C, existence
and uniqueness in generalized smoothness classes is derived.
61
Throughout the sequel,
represents the L evy process associated to the L evy measure
PA
, see (4.1.3).
For any L evy measure PA
and ¡ 0,
pq :
»
1
p{q pq,PℬpR
0
q, r
pq :pq
pq. (4.1.7)
For any L evy measure PA
, we denote its symmetrization
pq
1
2
r pq pqs.
AndA
t PA
:
u.
If
R
is a space of functions on R
with norm ||
||
pR
q
, then p
q
denotes the spaces of functions on
r0,s R
with nite norm
||
||
p
q
sup
Pr0,s
|p,q|
pR
q
.
4.2 Properties of O-RV Functions
Let us rst state a few results that are used in this part. Let PA
,
pq pq pt||¡uq¡ 0,
pq pq
1
, ¡ 0.
and lim
Ñ0
pq 0. We assume that
is an O-RV function at zero, i.e.,
1
pq lim
Ñ0
pq
1
pq
1
8, ¡ 0.
By Theorem 2 in [2], the following limits exist:
1
lim
Ñ0
log
1
pq
log
¤
1
lim
Ñ8
log
1
pq
log
. (4.2.1)
62
Lemma 4.2.1. Assume
is an O-RV function at zero.
a) Let ¡ 0 and ¡
1
. There is ¡ 0 so that
»
0
pq
¤
pq
, Pp0, 1s,
and lim
Ñ0
pq
0.
b) Let ¡ 0 and
1
. There is ¡ 0 so that
»
1
pq
¤
pq
, Pp0, 1s,
and lim
Ñ0
pq
8.
c) Let 0 and ¡
1
. There is ¡ 0 so that
»
0
pq
¤
pq
, Pp0, 1s,
and lim
Ñ0
pq
0.
d) Let 0 and
1
. There is ¡ 0 so that
»
1
pq
»
1
1
1
¤
pq
, Pp0, 1s,
and lim
Ñ0
pq
8.
Proof. The claims follow easily by Theorems 3, 4 in [2]. Because of the similarities, we will
prove c) only. Let 0 and ¡
1
. Then
lim
Ñ8
1
1
lim
Ñ0
pq
p
1
q
lim
Ñ0
1
p
1
q
lim
Ñ0
pq
pq
1
pq
8, ¡ 0.
63
Hence
1
, ¥ 1, is an O-RV function at innity with
lim
Ñ0
log
1
pq
log
1
¤
1
lim
Ñ8
log
1
pq
log
.
Then for Pp0, 1s,
»
0
pq
»
8
1
1
¤
pq
by Theorem 3 in [2], and lim
Ñ0
pq
0 according to Theorem 4 in [2].
Corollary 4.2.2. Assume
is an O-RV function at zero and
1
¡ 0. Let ¡ 1,
¡ 0. Then
8
¸
0
8.
Proof. Indeed,
8
¸
0
¤
»
8
0
¤
»
1
0
pq
8,
because, by Lemma 4.2.1a),
»
0
pq
¤pq
, Pr0, 1s.
We will need some L evy measure moment estimates.
Lemma 4.2.3. Let PA
, and
be an O-RV function at zero with
1
,
1
defined in
(4.2.1). Assume
0
1
¤
1
1 if Pp0, 1q,
1 ¤
1
¤
1
2 if 1,
1
1
¤
1
2 if Pp1, 2q.
64
Then
(i)
sup
Pp0,1s
»
p||^ 1qr
pq 8 if Pp0, 1q,
sup
Pp0,1s
»
||
2
^ 1
r
pq 8 if 1,
sup
Pp0,1s
»
||
2
^||
r
pq 8 if Pp1, 2q.
(ii)
inf
Pp0,1s
»
||¤1
||
2
r
pq¥
1
,
for some
1
¡ 0.
Proof. (i) Let Pp0, 1q. Then by Lemma 4.2.1,
»
||¤1
||r
pq
1
»
||¤
|| pq
1
»
0
r pq pqs,
and
»
||¤1
p||^ 1qr
pq
1
»
0
pq
1
¤, Pp0, 1s.
Let 1. Then, using Lemma 4.2.1 we have
»
||¤1
||
2
^ 1
r
pq 2
2
»
0
2
pq
1
¤, Pp0, 1s.
Let Pp1, 2q. Then similarly,
1
»
||¡
|| pq
1
»
8
0
p_q
pq
»
8
pq pq
»
8
pq
1
65
and with Pp0, 1s,
2
»
||¤
||
2
pq 2
2
»
0
2
rpq
1
pq
1
s
(4.2.2)
2
2
»
0
2
pq
1
pq
1
.
Hence, by Lemma 4.2.1,
»
||
2
^||
r
pq
¤ 2
2
»
0
2
pq
1
»
1
pq
1
»
8
1
pq
1
2
2
»
0
2
pq
1
»
1
pq
1
»
||¡1
|| pq
¤ pq
1
, Pp0, 1s.
(ii) By (4.2.2), for Pp0, 1s,
»
||¤1
||
2
r
pq pq
»
||¤1
||
2
pq
2
2
»
0
2
r
pq
pq
1s
2
»
1
0
2
r
pq
pq
1s
.
Hence, by Fatou's lemma,
lim
Ñ0
»
||¤1
||
2
r
pq¥ 2
»
1
0
2
r
1
1
pq
1s
1
¡ 0
if|tPr0, 1s :
1
pq 1u|¡ 0, and
lim inf
Ñ0
pq
pq
1
lim sup
Ñ0
pq
pq
1
1
pq
, Pp0, 1s.
66
According to [12], Chapter 3, 70-74, any L evy measure PA
can be disintegrated as
pq
»
8
0
»
1
Γ
pq p,q pq, Pℬ
R
0
,
where , and p,q,¡ 0, is a measurable family of measures on the unit sphere
1
with p,
1
q 1,¡ 0. The following is a straightforward consequence of Lemma
4.2.3(ii).
Corollary 4.2.4. Let PA
,
pq
»
8
0
»
1
Γ
pq p,q pq, Pℬ
R
0
,
where , p,q, ¡ 0, is a measurable family of measures on
1
with
p,
1
q 1, ¡ 0. Assume
1
be an O-RV function at zero satisfying
assumptions of Lemma 4.2.3, and
inf
|
p
|1
»
1
p
2
p,q¥
0
¡ 0. (4.2.3)
Then assumption B holds.
Proof. Indeed, for|
p
| 1,Pp0, 1s, with ¡ 0,
»
||¤1
p
2
pq
2
»
||¤
p
2
pq
2
»
0
»
1
p
2
p,q
2
pq
¥
0
2
»
0
2
pq
0
2
»
||¤
||
2
pq
0
»
||¤1
||
2
pq.
Hence by Lemma 4.2.3(ii),
inf
Pp0,1s
inf
|
p
|1
»
||¤1
p
2
r
pq ¥
0
inf
Pp0,1s
»
||¤1
||
2
r
pq¥
0
1
¡ 0.
67
Remark 4.2.5. Let P p0, 2q, P A
, and
be an O-RV function at zero,
1
¡ 0. By
Theorems 3 and 4 in [2], for any Pp0,
1
q,
»
||¤1
||
pq »
1
pq
1
p1q¥
pq
1
p1qÑ8
as Ñ 0. Hence
1
¤ . On the other hand for any ¡
1
, by Lemma 4.2.3,
»
0 ||¤1
||
pq¤ »
1
0
pq
1
8,
and ¤
1
.
68
Chapter 5
Norm Equivalence
We x a constant ¡ 1. For such an , by Lemma 6.1.7 in [6] and appropriate scaling,
there exists P
8
0
R
such that supppqt :
1
¤| |¤ u, p q¡ 0 in the interior
of its support, and
8
¸
8
1 if 0.
We denote throughout this paper
ℱ
1
rp
qs, 1, 2,..., P R
, (5.0.1)
0
ℱ
1
r1
8
¸
1
p
qs. (5.0.2)
Apparently,
P
R
,P N. They are convolution functions we use to dene generalized
Besov spaces. Namely, for ¡ 0 we write
r
8,8
R
as the set of functions in
1
R
for
which
||
, 8
: sup
|
|
0
8, (5.0.3)
where
with PA
.
Lemma 5.0.1. Let PA
,
be an O-RV function at zero and A holds for it. Let
¡ 0. If P
r
8,8
pR
q, then is bounded and continuous,
pq
8
¸
0
p
qpq, P R
,
where the series converges uniformly. Moreover,
||
0
¤||
, 8
, P
r
8,8
pR
q.
69
Proof. Note that
is continuous of moderate growth and
°
8
0
in
1
pR
q.
Obviously, by Corollary 4.2.2,
8
¸
0
|
|
0
8
¸
0
p
q
p
q
|
|
0
¤ sup
¥0
p
q
|
|
0
8
¸
0
p
q
¤ ||
, 8
8
¸
0
p
q
¤ ||
, 8
.
Let PA
,
, ¡ 0. For P
8
pR
q, set
||
0
sup
|pq|,
rs
sup
,ℎ0
|pℎqpq|
p|ℎ|q
,
and
||
:||
0
rs
.
Proposition 5.0.2. Let PA
,
be an O-RV function at zero so that A and C hold
for it. Let Pp0,
1
1
q. Then the norm ||
and norm ||
, 8
are equivalent on
8
pR
q.
Namely, there is ¡ 0 depending only on ,, such that
1
||
¤||
, 8
¤||
, P
8
pR
q.
Proof. Let P
8
pR
q. Then, by Lemma 4.2.1,||
8. If 0, then
| 0
|
0
¤||
0
»
| 0
pq|¤||
.
70
If 0, then by the construction of
,
³
pq p
p0q 0. Therefore, denoting
ℱ
1
,
|
|
0
»
rpqpqs
pq
0
¤ rs
»
p||q
pq
rs
»
||
| pq|.
Since for
0
¡ 1,
| pq|¤p1||q
0
, P R
for some ¡ 0, we have
»
||
| pq|
¤
»
1
0
||
»
8
1
||
||
0
1
2
.
By Lemma 4.2.1
p
0
q
¤p
q
, ¥ 0,
and
1
¤
»
0
pq
¤
,
2
p
0
q
»
8
pq
p
0
q
p
0
q
»
1
pq
p
0
q
p
0
q
»
8
1
pq
p
0
q
¤ p
q
, ¥ 0.
That is to say||
, 8
¤||
, P
pR
q for some constant p, q¡ 0.
71
Let
r
,
r
0
P
8
0
pR
q, be such that 0Rsuppp
r
q,
r
,
r
0
0
0
, where
0
ℱ 0
,
and , 0
are the functions introduced in (5.0.2), (5.0.1). Let
r ℱ
1
r
, r 0
ℱ
1
r
0
,
ℱ
1
r
p
q, ¥ 1.
Hence
r
, ¥ 0,
where in particular,
r
pq
r p
q, ¥ 1, P R
.
Obviously,
| 0
pq 0
pq| ¤
»
|r 0
pq r 0
pq|| 0
pq|
¤ p||^ 1q| 0
|
0
, ,P R
,
and
|
pq
pq|
¤
»
r
pq
r
pq
|
pq|
¤
^ 1
|
|
0
,¥ 1, ,P R
.
By Lemma 5.0.1, for ,P R
,
|pqpq| ¤
8
¸
0
|
pq
pq|
¤
8
¸
0
||^ 1
|
|
0
.
72
Let
1
1, P N. For||Pp
1
,
s,
|pqpq| ¤ ||
, 8
sup
||¤
8
¸
0
||^ 1
¤ ||
, 8
¸
0
8
¸
1
.
Then, by Lemma 4.2.1,
¸
0
¤
2
»
1
0
¤
»
1
1
1
pq
¤p||q
.
Again, by Lemma 4.2.1,
8
¸
1
¤
»
8
1
¤
»
1
0
pq
¤p||q
.
The statement is proved.
5.1 Equivalent Norms on
8
pR
q
Now we will introduce some other norms on
8
R
involving the powers of the operators
,
:
||
,,
||
,
||
0
|
; |
, 8
,P
8
R
,
||||
;,
||||
,
|p
q
|
, 8
,P
8
R
,
with , ¡ 0,
; p
q
, and satisfying A and B. In addition, we assume that
PA
t PA
:
u if is fractional. First, we dene those powers and corre-
sponding norms on
8
pR
q. Then we study their relations and extend them to
r
88
pR
q.
73
For PA
, Pp0, 1q, ¥ 0, and PpR
q, we see easily that
p p qq
p
p q
»
8
0
expp p qq 1
p
p q, P R
,
and dene
p
q
pq ℱ
1
p q
p
pq (5.1.1)
E
»
8
0
p
qpq
, P R
,
where
»
8
0
1
1
.
We denote, with 0, PpR
q, Pp0, 1q,
; :ℱ
1
rp q
p
s.
For P
8
pR
q, Pp0, 1q, ¥ 0, we dene
p
q
pq
E
»
8
0
p
qpq
, P R
.
For 1,p
q
1
p
q
, P
8
pR
q. Note that for Pp0, 1q,¥
0,
p
q
pq
E
»
8
1
p
qpq
(5.1.2)
E
»
1
0
»
0
p
qp
q
,
P R
.
For ¡ 0, P
8
pR
q, set
p
q
pq
1
»
8
0
Ep
q
, P R
,
74
where
1
»
8
0
1
,
and PA
, ¡ 0, or PA
, P N.
Note that for P
R
,
ℱ
p
q
p q
p , ¡ 0, ¡ 0,
ℱrp
q
s p q
p , ¥ 0, Pp0, 1s.
We use the formulas above to denep
q
,¥ 0, 1, 0,1,..., for PA
.
Remark 5.1.1. Assume Pp0, 1s,¥ 0 or Pp8, 0q,¡ 0. It is easy to see that
a) for any P
8
R
, we have p
q
P
8
R
and for any multiindex
,
p
q
p
q
, P A
. The same holds for P A
and
1, 0,1,....
b) for any P
8
R
such that for any multiindex ,
P
1
R
X
2
R
, we
have
ℱ
p
q
p q
p
, ¡ 0, ¡ 0,
ℱrp
q
s p q
p
, ¥ 0, Pp0, 1s,
for PA
. The same holds for PA
and 1, 0,1,....
The following obvious claim holds.
Lemma 5.1.2. Let P A
. Assume P p0, 1s, ¥ 0 or P p8, 0q, ¡ 0. Let
,
P
8
R
be so that for any multiindex ,
Ñ
as Ñ 8 uniformly on
compact subsets of R
and
sup
,
|
pq| 8.
75
Then for any multiindex ,
p
q
p
q
Ñ
p
q
p
q
uniformly on compact subsets of R
, and
sup
,
|p
q
pq| 8.
The same holds for PA
and 1, 0,1,....
Remark 5.1.3. Given P
8
R
there is a sequence
P
8
0
R
so that for any
multiindex ,
Ñ
as Ñ8 uniformly on compact subsets of R
and
sup
,
|
pq| 8.
Indeed, choose P
8
R
, 0¤ ¤ 1,pq 1 if||¤ 1, and pq 0 if||¡ 2.
Given P
8
R
, take
pqpqp{q, P R
, ¥ 1.
Lemma 5.1.4. Let PA
. Assume ¡ 0, Pp0, 1s. Then p
q
:
8
R
Ñ
8
R
is bijective whose inverse isp
q
:
p
q
p
q
pqp
q
p
q
pqpq, P R
,
for any P
8
R
.
Proof. It is an easy consequence of Lemma 5.1.2 and Remarks 5.1.1 and 5.1.3.
For an integer P N, we dene for PA
,
p
q
p
q...p
q
loooooooooooooomoooooooooooooon
times
.
76
For a non integer ¡ 0, r s with Pp0, 1q and PA
, we set
p
q
p
q
r s
p
q
p
q
p
q
r s
, P
8
pR
q.
Remark 5.1.5. Let PA
, and P
8
pR
q be such that for any multiindex ,
P
1
pR
qX
2
pR
q. Then
ℱ
p
q
p q
p
, ¡ 0, ¡ 0,
ℱrp
q
s p q
p
, ¥ 0, ¡ 0.
The same holds with PA
if P N.
Lemma 5.1.6. Assume ¡ 0. Then
(i) for any , ¥ 0, we have
;
;
;
; for any , P R,
p
q
p
q
p
q
,
p
q
p
q
p
q
p q
,
for PA
.
The same holds with PA
if , P N.
(ii) for any ¡ 0, the mapping p
q
:
8
pR
q Ñ
8
pR
q is bijective whose
inverse isp
q
:
p
q
p
q
pqp
q
p
q
pqpq, P R
.
for any P
8
pR
q.
The same holds with PA
if P N.
Proof. The statement is an easy consequence of Lemma 5.1.2 and Remarks 5.1.1, 5.1.3, and
5.1.5.
77
Lemma 5.1.7. Let PA
satisfy A.
(i) Let ¥ 0, ¡ 0,r s 1. For any P
8
pR
q,
sup
Pp0,1s,
r
pq
¤p1q
max
| |¤| |2
|
|
0
8.
If, in addition, for any multiindex ,
³
|
pq| 8, then
sup
Pp0,1s,
»
r
pq
¤p1q
max
| |¤| |2
»
|
pq|.
The same holds with PA
satisfying A if P N.
(ii) Let ¡ 0, ¡ 0. For any P
8
pR
q,
sup
Pp0,1s,
r
pq
¤
max
| |¤| |
|
|
0
8.
If in addition, for any multiindex ,
³
|
pq| 8, then
sup
Pp0,1s
»
r
pq
¤
max
| |¤| |
»
|
pq|.
The same holds with PA
satisfying A if P N.
Proof. (i) Let Pp0, 1s. Then
r
pq
»
8
0
E
r
pq
»
8
1
...
»
1
0
»
0
E
p
r
q
r
,
P R
.
By Lemma 4.2.3, we have
sup
Pp0,1s
»
p||^ 1qr
pq 8 if Pp0, 1q,
78
sup
Pp0,1s
»
||
2
^ 1
r
pq 8 if 1,
sup
Pp0,1s
»
||
2
^||
r
pq 8 if Pp1, 2q,
and both inequalities easily follow. Applying them repeatedly we obtain the claim for an
arbitrary ¡ 0.
(ii) Indeed, for any ¡ 0,¡ 0, and any multiindex ,
r
pq
»
8
0
E
r
, P R
,
and the claim obviously follows.
Lemma 5.1.8. Let P A
satisfy A and B. Let P
R
be such that p P
8
0
R
, 0Rsupppp q. Then there are constants , so that
sup
Pp0,1s
»
E
r
¤
, ¡ 0.
Proof. Let p,q Ep
r
q, P R
, ¡ 0. We choose ¡ 0 so that supp pp q
:| |¤
1
(
. Let r ,
pq t||¤u
r
pq, Pr0, 1s. Then for Psupppp q and||¤,
1 cosp q¥
1
| |
2
| |
2
p
2
with
p
{| |. Therefore there is
0
¡ 0 so that for any Psupp pp q and Pp0, 1s,
r ,
p q
»
||¤
r1 cosp qsr
pq (5.1.3)
¥
| |
2
»
||¤
p
2
r
pq¥
0
| |
2
.
Then
p
p, q exp
!
r
p q
)
p p q exp
!
r ,
p q
)
expt p qup p q, P R
,
79
where expt p qu is a characteristic function of a probability distribution
,
pq on R
.
Hence
p,q
»
p,q
,
pq, P R
,
with
p,qℱ
1
rexpt r ,
up s Ep
r ,
q, P R
.
Since
»
|p,q|¤
»
|p,q|,
it is enough to prove that
»
|p,q|¤
, ¡ 0. (5.1.4)
Now, (5.1.3) implies that for any multiindex| |¤r{2s 3,
»
|
p,q|
2
¤
»
p p q exp
!
r ,
p q
)
2
¤
1
2
, ¡ 0.
Hence, denoting
0
r{2s 1,
»
|p,q|
»
1||
2
0
|p,q|
1||
2
0
¤
»
|p,q|
2
1||
2
2
0
¤
1
2
, ¡ 0.
Thus (5.1.4) follows, and
»
E
r
¤
1
2
, ¡ 0. (5.1.5)
80
Corollary 5.1.9. Let P A
satisfy A and B. Let P pR
q be such that p P
8
0
pR
q, 0Rsupppp q. Then for ¥ 0, ¡ 0, Pp0, 1s,
p
r
q
pq ℱ
1
rp r
q
p spq
»
8
0
Ep
r
q
, P R
,
is
8
pR
q-function, and for every multi-index , we have
p
r
q
p
r
q
, ¡ 0, and
sup
Pp0,1s,¥0
»
|
p
r
q
pq|
8, ¥ 1.
The same holds with PA
satisfying A and B if P N.
Proof. Take P
8
0
pR
q so that p p , 0Rsuppp q, and let r ℱ
1
. Let
p,q Ep
r
q, ¡ 0, P R
.
Then
p
p, q expt r
p qu p qp p q, P R
,
and
p,q
»
p,qpq
»
pq
p,q, P R
,
with
p,qℱ
1
rexpt r
u s Er p
r
q, ¡ 0, P R
.
By Lemma 5.1.8,
sup
Pp0,1s
»
|
p,q|¤
,¡ 0.
Hence
p,qP
8
pR
q,¡ 0, and for each multi-index and ¥ 1,
sup
»
|
p,q|
1{
¤
»
|
pq|
1{
, ¡ 0.
81
Corollary 5.1.10. Let P A
satisfy A and B. Let P pR
q be such that p P
8
0
pR
q, 0Rsupppp q. Then there are constants ,¡ 0 so that
sup
Pp0,1s
»
|E
r
p
r
q|¤
, ¡ 0.
Proof. LetℎP
8
0
pR
q, 0¤ℎ¤ 1, andℎp q 1 if P supppq,ℎp q 0 in a neighborhood
of zero. Let
p,q E
r
p
r
q, P R
.
Then
p
p, q expt r
p qu r
p qp p q expt r
p quℎp q r
p qp p q, P R
.
Hence
p,q
»
p,q
pq, P R
,
where
pq
r
pq,
p,q Eℎp
r
q, P R
.
Thus, by Lemma 5.1.8
sup
Pp0,1s
»
|
p,q| ¤ sup
Pp0,1s
»
|
p,q| sup
Pp0,1s
»
|
pq|
¤
, ¡ 0.
Lemma 5.1.11. Let PA
satisfy A and B. Then
(i) For each , ¡ 0, there is ¡ 0 so that
|
; |
, 8
¤ ||
, 8
, P
8
pR
q,
82
|p
q
|
, 8
¤ ||
, 8
, P
8
pR
q,
(ii) For each 0 , there is ¡ 0 so that
||
, 8
¤
|
; |
, 8
||
0
, P
8
pR
q,
||
, 8
¤ |p
q
|
, 8
, P
8
pR
q.
The same holds with PA
satisfying A and B if P N.
Proof. Let P
8
pR
q,
r
,
r
0
P
8
0
pR
q, be such that
r
,
r
0
0
0
, where
0
ℱ
1
0
, and , 0
are the functions in the denition of the spaces.
(i) Let Pr0, 1s. Then
p
q
ℱ
1
p q
r
p
»
pq
pq, P R
, ¥ 1,
p
q
0
ℱ
1
p q
r
0
p
0
»
0
pq 0
pq, P R
,
where
ℱ
1
p q
r
,¥ 1,
0
ℱ
1
p q
r
0
. Let
ℱ
1
r
r
, ¥ 1.
Since
p p qq
r
r
r
s
r
, P R
,
it follows by Lemma 5.1.7 that
»
pq
»
|
pq|
»
r
r pq
¤
, ¥ 0.
83
(ii) Let 0 , Pr0, 1s. Then for ¥ 1,
p
q
p
q
ℱ
1
p q
r
p q
p
»
pqp
q
pq, P R
, ¥ 1,
where
ℱ
1
p q
r
, ¥ 1, ¥ 0.
Let
ℱ
1
r
r
, ¥ 1.
It follows by Corollary 5.1.9 that there is independent of ¥ 0, ¥ 1, so that
»
pq
»
|
pq|
»
r
r pq
¤
.
On the other hand,
| 0
|
0
¤||
0
.
The statement follows.
For ¡ 0, ¡ 0, we dene the following norms:
||
,,
||
0
|
; |
, 8
,P
8
pR
q,
||||
;,
|p
q
|
, 8
,P
8
pR
q,
with satisfying A and B. An immediate consequence of Lemma 5.1.11 is the following
norm equivalence.
84
Corollary 5.1.12. Let P A
be a L´ evy measure satisfying A and B, ¡ 0, ¡ 0.
Then norms||
,,
,}}
,,
and||
, 8
are equivalent on
8
pR
q.
The same holds with PA
satisfying A and B if P N.
Proof. Let , ¡ 0. By Lemma 5.1.11,
|p
q
|
, 8
¤ ||
, 8
¤
|
; |
, 8
||
0
||
;,
, P
8
pR
q.
On the other hand, by Lemmas 5.0.1 and 5.1.11,
||
0
¤ ||
, 8
¤||
, 8
¤|p
q
|
, 8
,
|
; |
, 8
¤ ||
, 8
¤|p
q
|
, 8
, P
8
pR
q.
Corollary 5.1.13. Let PA
and PA
be a L´ evy measure satisfying A and B such
that
. Then for any P N, ¡ 0, there are constants ,¡ 0 so that
|p
q
|
, 8
¤
1
||
;,
¤
2
|p
q
|
, 8
||
0
, P
8
pR
q.
Proof. Indeed, by Corollary 5.1.12,
|p
q
|
, 8
¤ ||
, 8
¤||
;,
¤ ||
, 8
¤
2
|p
q
|
, 8
||
0
, P
8
pR
q.
85
5.2 Extension of Norm Equivalence to
r
88
pR
q
We extend the denition ofp
q
and the norm equivalence (see Corollary 5.1.12 above)
from
8
pR
q to
r
88
pR
q. We start with the following observation.
Remark 5.2.1. Let 0 1
. Then for each ¡ 0 there is a constant
¡ 0 so that
||
1
,8
¤||
, 8
||
0
,P
r
88
pR
q.
Indeed, For each ¡ 0 there is ¡ 1 so that p
q
1
¤ if ¥. Hence
1
|
|
0
1
|
|
0
¤ ||
, 8
max
¤
1
|
|
0
¤ ||
, 8
||
0
.
Proposition 5.2.2. Let ¡ 0, P
r
8,8
pR
q. Then there exists a sequence
P
8
pR
q,
such that
||
, 8
¤ lim inf
|
|
, 8
, |
|
, 8
¤||
, 8
for some ¡ 0 that only depends on ,. Moreover, for any 0 1
,
|
|
1
,8
Ñ 0 as Ñ8.
Proof. Set
°
0
, ¥ 1. Obviously,
P
8
pR
q, ¥ 1, and by Lemma 5.0.1,
°
8
0
is a bounded continuous function. Since
1
¸
1
,¥ 1, 0
p 0
1
q 0
,
we have for ¡ 1,
p
q
0, ,
86
p
q
p 1
1
q
, ¡ 1,
p
q
p 1
q
,
p
q 1
p 1
2
q 1
.
Hence there is a constant so that
|
|
0
¤|
|
0
, ¥ 0, ¡ 1,
and for ¡ 1,
sup
|
|
0
sup
|
|
0
¤|
|
, 8
.
Thus
||
, 8
¤ lim
|
|
, 8
,
and
|
|
, 8
¤||
, 8
, ¡ 1.
Now, by Remark 5.2.1, for each ¡ 0, there is a constant
so that
|
|
1
,8
¤
|
|
, 8
||
, 8
|
|
0
.
Since by Lemma 5.0.1,|
|
0
Ñ 0, the statement follows.
Using the approximating sequence introduced in Proposition 5.2.2, we can extend
; ,p
q
, 0 , to all P
r
8,8
pR
q, ¡ 0.
Proposition 5.2.3. Let be a L´ evy measure satisfyingA andB, ¡ 0 andP
r
8,8
pR
q.
Let
P
8
pR
q be an approximating sequence of in Proposition 5.2.2. Then for each
87
Pp0, q there are bounded continuous functions, denoted p
q
,
; P
r
88
, so
that for any 0 1
,
|
;
; |
1
,8
|p
q
p
q
|
1
,8
Ñ 0
as Ñ8. Moreover, for each Pp0, q there is ¡ 0 independent of P
r
8,8
pR
q so
that
|
; |
, 8
¤||
, 8
, |p
q
|
, 8
¤||
, 8
, (5.2.1)
and
||
, 8
¤
|
; |
, 8
||
0
, ||
, 8
¤|p
q
|
, 8
. (5.2.2)
Proof. Let P
r
8,8
pR
q. By Proposition 5.2.2, there is a a sequence
P
8
pR
q such
that
||
, 8
¤ lim inf
|
|
, 8
, |
|
, 8
¤||
, 8
,¥ 1,
for some ¡ 0 independent of , and for any , 1
Pp0, q, see Lemma 5.1.11 as
well,
|
;
;
|
1
,8
|p
q
p
q
|
1
,8
¤ |
|
1Ñ 0 as ,Ñ8.
Hence there are bounded continuous functions, denoted
; ,p
q
, so that
|
;
; |
0
|p
q
p
q
|
0
Ñ 0
as Ñ8. Thus
|
;
;
|
0
|p
q
p
q
|
0
Ñ 0, ¥ 0
88
as Ñ8. Now, for each ¡ 1, and 0, 1,
sup
¤
|p
q
|
0
lim
Ñ8
sup
¤
|p
q
|
0
¤ sup
|p
q
|
¤ sup
|
|
, 8
¤||
, 8
.
Hencep
q
P
r
88
R
, 0, 1, and (5.2.1) holds.
Now for every ¥ 0, we have
rp
q
s
lim
rp
q
s
p
q
r
s (5.2.3)
uniformly. By the denition of the approximation sequence (see proof of Proposition 5.2.2),
|p
q
r
s|
0
¤|p
q
r
s|
0
,¥ 0.
Hence
||
, 8
¤ lim inf
|
|
, 8
¤ lim inf
|p
q
|
, 8
¤|p
q
|
, 8
,
and similarly,
||
, 8
¤ lim inf
|
|
, 8
¤ lim inf
|
;
|
, 8
|
|
0
¤
|
; |
, 8
||
0
.
Proposition 5.2.4. Let PA
be a L´ evy measure satisfying A and B, ¡ 0, ¡ 0.
Then norms||
,,
,}}
,,
and||
, 8
are equivalent on
r
88
pR
q.
Proof. We show the equivalence by repeating proof of Corollary 5.1.12 where the equiva-
lence of the same norms on
8
pR
q was derived. Only instead of Lemma 5.1.11 we use
Proposition 5.2.3.
89
Chapter 6
Proof of Main Theorem
We assume in this chapter that A , B and C hold. First we solve the equation with smooth
input functions.
Proposition 6.0.1. Let PA
, Pp0, 1q, ¥ 0. Assume that p,qP
8
p
q. Then
there is a unique solution P
8
p
q to
B
p,q
p,q p,qp,q, (6.0.1)
p0,q 0, p,qPr0,s R
.
Proof. Existence. Denote p,
q
pq
p,
q,¤ ¤ , and apply
the It^ o formula to p,
q onr,s.
pq
p,
qp,q
»
p,
q
»
»
pq∇
,
r
p,q
»
»
,
,
pq∇
,
p,q.
Take expectation for both sides and use the stochastic Fubini theorem,
pq
Ep,
qp,q
»
pq
Ep,
q
»
pq
Ep,
q.
Integrate both sides overr0,s with respect to and obtain
»
0
pq
Ep,
q
»
0
p,q
90
»
0
»
0
pq
Ep,
q
»
0
»
0
pq
Ep,
q,
which shows p,q
³
0
pq
E
,
solves (6.0.1) in the integral sense.
Obviously, as a result of the dominated convergence theorem and Fubini's theorem, P
8
p
q. And by the equation, is continuously dierentiable in .
Uniqueness. Suppose there are two solutions
1
,
2
solving the equation, then :
1
2
solves
B
p,q
p,q p,q, (6.0.2)
p0,q 0.
Fix anyPr0,s. Apply the It^ o formula top,
q :
p,
q, 0¤¤,
overr0,s and take expectation for both sides of the resulting identity, then
p,qE
»
0
pB
q
,
0.
6.1 H¨ older Estimates of the Smooth Solution
First we derive the estimates of the solution corresponding to a smooth input function.
Proposition 6.1.1. Let P A
, ¡ 0 and A-C hold. Let P
8
p
q be the unique
solution to (6.0.1) with P
8
p
q. Then
||
, 8
¤ pq||
, 8
, (6.1.1)
||
1, 8
¤ r1 pqs ||
, 8
(6.1.2)
91
and for any Pr0, 1s,
1
¤ ,
p,q
1
,
, 8
(6.1.3)
¤
!
1
1 r1 pqs
1
)
||
, 8
,
where pq
1
^.
Proof. Since P
8
p
q, by Lemma 5.0.1,
p,q pp,q 0
pqqpq
8
¸
1
pp,q
pqqpq
0
p,q
8
¸
1
p,q, p,qP
.
Accordingly, for ¥ 0,
p,qp,q
pq
»
0
pq
E
,
, p,qP
,
is the solution to (6.0.1) with input
. In terms of Fourier transform,
p
p, q
»
0
exptp p qqpqu
p
p, q
»
0
pq
exp
!
r
1
pq
)
r
p
p, q, ¥ 1.
Denote
1
. Then for ¥ 0,
p,q
»
0
pq
»
p,q
p,q,Pr0,s, P R
,
92
with
p,q
Er
r
, p,qP
, ¥ 1,
0
p,q Er 0
p
q, p,qP
.
Hence
»
|
p,q|
»
|
p,q|, ¡ 0, ¥ 0, (6.1.4)
with
0
0
and
p,q Er
r
, p,qP R
, ¥ 1.
First we estimate the solution itself. For ¥ 1, by Lemma 5.1.8,
|
p,q|
0
¤ |
|
0
»
0
pq
»
p,q
¤ |
|
0
»
0
pq
pq
¤
1
|
|
0
.
Directly,
|
0
p,q|
0
¤|
0
|
0
»
0
pq
¤
1
^
|
0
|
0
.
Hence
||
1, 8
¤
1
1
^
||
, 8
.
Now we estimate time dierences. For xed 0
1
¤,¥ 0,
p,q
1
,
»
1
pq
»
p,q
p,q
p
1
q
1
»
1
0
p
1
q
»
p,q
p,q
»
1
0
p
1
q
»
p,q
1
,
p,q
93
1
pq
2
pq
3
pq, P R
.
First, by Lemma 5.1.8, for ¥ 1,
1
0
¤
»
1
pq
»
p,q
|
|
0
¤
»
1
pq
pq
|
|
0
¤
»
1
pq
|
|
0
¤
1
1
p
1
q
|
|
0
.
And
0
1
¤|
0
|
0
»
1
pq
¤|
|
0
1
.
By (6.1.4) and Lemma 5.1.8, for ¥ 1,
2
0
¤
1
p
1
q
»
1
0
p
1
q
»
p,q
|
|
0
¤
1
p
1
q
»
1
0
p
1
q
pq
|
|
0
.
Thus for ¥ 1,
2
0
¤
1
p
1
q
»
1
0
p
1
q
|
|
0
¤|
|
0
1
, (6.1.5)
in the mean time,
2
0
¤|
|
0
»
1
0
p
1
q
¤|
|
0
1
. (6.1.6)
For 0,
0
2
0
¤
1
p
1
q
»
1
0
p
1
q
|
|
0
¤
1
»
1
0
p
1
q
|
|
0
¤ |
|
0
1
.
94
At last, for ¥ 1,
3
0
¤|
|
0
»
1
0
»
p,q
1
,
.
Note for ¤
1
,
p,q
1
,
E
r
r
pq
r
r
p
1
q
E
»
pq
p
1
q
r
r
r
,
and by Corollary 5.1.10,
»
p,q
1
,
¤
»
pq
p
1
q
¤
p
1
q
1
p
1
q
.
Thus for ¥ 1,
3
0
¤ |
|
0
1
p
1
q
»
1
0
p
1
q
1
|
|
0
1
p
1
q
1
1
¤ |
|
0
1
1
p
1
q
.
In addition,
0
3
0
¤ |
0
|
0
»
1
0
p
1
q
1
¤
1
^
|
0
|
0
1
.
Summarizing,
0
p,q
0
1
,
0
¤
1
1
^
|
0
|
0
1
,
95
and
p,q
1
,
0
¤ |
|
0
1
^
1
1
1
p
1
q
|
|
0
1
1
^ 1
1
p
1
q
,
which leads to
p,q
1
,
0
¤
1
1 , Pr0, 1s, ¥ 1.
Thus
p,q
1
,
, 8
¤||
, 8
"
1
1
1
1
^
1
*
for any Pr0, 1s. The statement is proved.
6.2 General H¨ older Inputs
Existence and Estimates. Given P
r
8,8
p
q, by Proposition 5.2.2, we can nd a
sequence of functions
in
8
p
q such that
|
|
, 8
¤||
, 8
, ||
, 8
¤ lim inf
|
|
, 8
,
and for any 0 1
,
|
|
0
¤|
|
1
,8
Ñ 0 as Ñ8.
According to Theorems 6.0.1 and 6.1.1, for each
P
8
R
, there is a corresponding
solution
P
8
p
q :
p,q
»
0
r
p,q
p,q
p,qs, p,qPr0,s R
. (6.2.1)
96
By Theorem 6.1.1,
|
|
1
,8
¤ |
|
1
,8
¤|
|
1 1
,8
¤ |
|
1
,8
Ñ 0, as ,Ñ8
for all 1
Pp0, q, which by Lemma 5.0.1 implies that
|
|
0
|
|
0
Ñ 0 as ,Ñ8.
So, there is P
r
1 1
8,8
p
q for any 1
Pp0, q such that|
|
1 1
,8
Ñ 0 as Ñ8.
Passing to the limit in (6.2.1) we see that (6.2.1) holds for. Let 1
Pp0, q and 1
1
1
.
Then
1 1
1
,8
¤|
|
1, 8
¤|
|
, 8
¤||
, 8
implies that
1 1
p,q
1 1
p,q
¤||
, 8
p||q
1
, ,P R
.
and passing to the limit we see that
1 1
p,q
1 1
p,q
¤||
, 8
p||q
1
, ,P R
.
Hence
1 1
P
r
1
8,8
pR
q, i.e., P
r
1 8,8
p
q and
||
1, 8
¤||
, 8
.
The convergence of
to implies easily other estimates.
97
Uniqueness. Suppose there are two solutions
1
,
2
P
r
1 8,8
p
q to (4.1.2), then
:
1
2
solves
p,q
»
0
r
p,q p,qs, p,qPr0,s R
. (6.2.2)
Let P
8
0
pR
q, 0¤¤ 1,
³
1. For ¡ 0, set
p,q
»
p,q
pq
»
p,q
pq, p,qP
,
with
pq
p{q, P R
. Then
P
r
8
p
q solves (6.2.2). Hence
0 for all
¡ 0. Thus 0, the solution is unique.
98
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Asset Metadata
Creator
Xu, Fanhui (author)
Core Title
On the non-degenerate parabolic Kolmogorov integro-differential equation and its applications
Contributor
Electronically uploaded by the author
(provenance)
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Mathematics
Publication Date
05/08/2019
Defense Date
03/18/2019
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Holder regularity,Levy process,OAI-PMH Harvest,parabolic integro-differential equation,stochastic differential equation,strong solution
Format
application/pdf
(imt)
Language
English
Advisor
Mikulevicius, Remigijus (
committee chair
), Ghanem, Roger (
committee member
), Kukavica, Igor (
committee member
)
Creator Email
fanhui.xu.math@gmail.com,fanhuixu@usc.edu
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https://doi.org/10.25549/usctheses-c89-168943
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Abstract (if available)
Abstract
This dissertation consists of two parts. In Part I, a stochastic differential equation (SDE) driven by an α-stable process with αε[1,2) is considered. It is assumed that this SDE has a Lipschitz jump coefficient and a β-Holder continuous drift with βε (0,1]. The existence and uniqueness of a pathwise solution is proved for β>1-α/2 by showing that it is the Lᵖ-limit of Euler approximations. When βε (0,1), the Lᵖ-error (rate of convergence) is obtained for non-degenerate truncated and non-truncated driving processes. When β = 1, the rate of convergence is derived for possibly degenerate truncated and non-truncated driving processes. In the case where β<1, regularity properties of the solution to a Kolmogorov integro-differential equation are used to overcome the difficulty brought by the rough jump coefficient. ❧ In Part II, a Cauchy problem for a non-degenerate parabolic Kolmogorov integro-differential equation is investigated. The primary operator of this equation is an infinitesimal generator of a Levy process which has a radially O-regularly varying Levy measure. Function spaces of generalized Holder smoothness are defined using the Levy measure. Existence and uniqueness of a strong solution, together with its space and time regularity are established in generalized Holder spaces.
Tags
Holder regularity
Levy process
parabolic integro-differential equation
stochastic differential equation
strong solution
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