Kobe University Repository : Thesis学位論文題目Tit le

Asymptot ic expansions for funct ionals of a Poissonrandom measure(Poisson配置の汎関数に対する漸近展開定理)

氏名Author Hayashi, Masafumi

専攻分野Degree 博士（理学）

学位授与の日付Date of Degree 2008-03-25

資源タイプResource Type Thesis or Dissertat ion / 学位論文

報告番号Report Number 甲4161

権利Rights

URL http://www.lib.kobe-u.ac.jp/handle_kernel/D1004161

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Create Date: 2017-12-20

Asymptotic expansions for functionals of a Poisson random measure

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Asymptotic expansions for functionals of a Poisson random measure

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Contents

1 Introduction 2

2 Notation 8 2.1 Probability space and hypothesis 8 2.2 Maps c+, c-, and operators D, D* 9 2.3 Spaces of random variables . 11

3 Preliminaries 16 3.1 Remarks on the domain of the operator D* 16 3.2 Smoothness criterion of Picard . . . . . . . . 19

4 Composites of functionals of a Poisson random measure and Schwarz distributions 24

5 Asymptotic expansions 5.1 Asymptotic expansions ..... 5.2 Asymptotic expansion theorem

6 Proof of Theorem 5.2.1

7 Applications to stochastic differential equations

8 Appendix 8.1 Closability of D 8.2 Proof of Theorem 3.2.3

28 28 31

34

42

56 56 57

Chapter 1

Introduction

In the Wiener space, asymptotic expansion theorem was obtained by Watan­abe [18]. The purpose of this paper is to prove this theorem for functionals of a Poisson random measure. To roughly explain this theorem, let us con­sider a family of Wiener functionals {F(E)j E E (0, I]} and some expansion F(E) rv ~::=O En In where In's are Wiener functionals. The series ~::=O En In does not necessarily converge. However, for each m, the finite sum ~:=O En In approximates the random variable F( E) in the following sense

m

I F(E) - LEn In I :s; Ck,p,mEm+I, n=O k,p

where the norm is the Sobolev norm in the sense of Malliavin calculus on the Wiener space. In the Malliavin calculus on Wiener space, for a Schwartz distribution T and for a smooth Wiener functional F(f.), one can define the composition T 0 F as a linear continuous functional on the space of smooth Wiener functionals (see [18]). The asymptotic expansion theorem asserts that, under some non-degenerate condition, the composition To F(f.) has also the asymptotic expansion ~::=o f.n4>n, and the expansion is given by the formal Taylor expansion. In [17], the author gave the asymptotic expansion F(f.) = X€2 where Xt is a solution to some stochastic differential equation driven by a Brownian motion. By applying asymptotic expansion theorem, one can estimate the short time behavior of the fundamental solution to the heat equation. One can also find applications of asymptotic expansion theorem in mathematical finance. We shall mention this application later.

The Malliavin calculus for functionals of a Poisson random measure has been developed by many authors. However, there are several methods of

2

introducing differential operators on the Poisson space. In fact, we have to choose an appropriate way as to the intensity measure of a Poisson ran­dom measure. Bismut [2] has generalized the Malliavin calculus for Wiener­Poisson functionals by using the Girsanov theorem. As another method, in Bichteler, Gravereaux and Jacod [1], one can find the study of the Malliavin operator on Wiener-Poisson space and application of it to stochastic differ­ential equations. Both in these works, the authors have given differential operators on Wiener-Poisson space and have proved the integration by parts formulas. These formulation suffers some limitation on an intensity measure, that is, the intensity measure must have a smooth density.

On the other hand, in the Malliavin calculus for Wiener functionals, Wiener chaos expansion of the space of square integrable Wiener functionals can be considered as a Fock space, and the differential operator is regarded as the annihilation operator on a Fock space. This sort of structure can be also found in the case of the space of square integrable functionals of Wiener-Poisson space, see [6]. Nualart and Vives [10], [11], and Picard [13] have studied the annihilation operator and its dual operator (the creation operator) on the space of square integrable functionals of a Poisson random measure. Picard [12] has also given a smoothness criterion by using the dual­ity formula (see Theorem 2.2.1) for functionals of a Poisson random measure under the Condition 1 (see Section 2.1) on the intensity measure, and has studied the solution to some stochastic differential equation. This argument of Picard can be generalized for some Wiener-Poisson functionals, see [5]. The Condition 1 differs from that of [1], and allows us to take an intensity measure with some singularity. One can find some interesting examples sat­isfying Condition 1, for instance, stable processes and CGMY processes (see [3]).

By using the Malliavin operator which we mentioned above, Sakamoto and Yoshida [15] have studied asymptotic expansion formulas of some Wiener­Poisson functionals in the statistical view point. In particular, the integration by parts formula has played an important role in [15]. However, as we men­tioned above, the intensity measure must have a smooth density. On the other hand, we adopt the framework of Picard [12]. Hence, we can consider functionals of a Poisson random measure with an intensity measure which may have some singularity. Let us roughly explain our main result (Theorem 5.2.1). We shall introduce Sobolev space Dk,p with norm i . ik,p in Section 2.3, and give a modification of smoothness criterion of Picard [12] in Section 3.2. If F E Doo := n~o np>2 Dk,p satisfies the non-degenerate condition, this

3

modification claims that

sup I E[GeiE .F]I :::; Ck ,p(1 + I e 12r(1-2p)~, IGlk,p=l

where c\(, {3 are some positive constants with I < {3. From this inequality, one can show that the function '¢(e) := E[GeiE .F] is a rapidly decreasing function, see Remark 2. In the Malliavin calculus on Wiener space, composites of Schwartz distributions and smooth Wiener functionals are linear continuous functionals on the space of smooth functionals, and can be evaluated by using the integration by parts formula. On the other hand, as we mentioned above, we cannot use the integration by parts formula in our formulation. Hence, to define composites of Schwatz distributions and functionals of a Poisson random measure as linear continuous functionals on D oo , we choose the following way; since '¢(e) = E[GeiE 'F ] is a rapidly decreasing function, we evaluate the composition as follows

(T 0 F, G) = s,(:FT, '¢ )s,

where T is a Schwartz distribution and :FT is the Fourier transform of T. In Chapter 4, we shall precisely discuss the definition. Now, we shall mention our main result. We shall consider the parametrized functionals F(f) f E

(0,1]. The asymptotic expansion F(f) f'V 2:~=o fn In can be defined similarly as that of Watanabe [18]. If F( f) has the asymptotic expansion and satisfies the uniformly non-degenerate condition, then the composition To F(f) has also the asymptotic expansion 2:~=o fn<I>n, where <I>n's are linear continuous functionals on Doc, and are given by the formal Taylor expansion. Hence, roughly speaking, the asymptotic expansion theorem for functionals of a Poisson random measure can be obtained similarly as that of Watanabe [18].

As an application, we shall give the asymptotic expansion of some stochas­tic differential equation. Our application is the analogue to the study of Kunitomo-Takahashi [7]. Kunitomo-Takahashi [7] [8] have applied the asymp­totic expansion of [17] and of [20], [19] to mathematical finance. In [9], they have considered the following stochastic differential equation:

where Ws is a Brownian motion, and have given the asymptotic expansion St( f) f'V 2:~o fj Aj . By using this expansion, they have estimated the option

4

price such as E[(St(€) - K)+]. The authors have also studied jump diffusion case:

where N is the compensated Poisson random measure whose intensity mea­sure is the Lebesgue measure or the normal distribution by using the formu­lation of [1]. On the other hand, we shall consider the following stochastic differential equation:

dXt(€) = b(Xt(€))dt + € L a(Xt-, y)N(dt x dy),

where the intensity measure of N satisfies the Conditionl. If the stock price process is given by the stochastic differential equation driven by a Levy pro­cess, to study the asymptotic expansion of X t (€) rv Ej:o €j!; seems impor­tant to evaluate the option price, as in [9]. However, we do not deal with numerical simulation in this work. Financial interpretation of the applica­tion can be found in [7],[8],[9]. As an example, we shall see that geometrical CGMY process satisfies the uniformly non-degenerate condition, although we need some modification on the tail of the density of the Levy measure.

The reminder of the paper is organized as follows: In the next chapter, we shall give general notation and introduce Sobolev spaces. In Chapter 3, we shall exhibit the preliminary results. In particular, we shall give a modification of the smoothness criterion of Picard for our purpose. Although the proof is essentially due to Picard , we need to give the modification of the proof of his main theorem, to prove the asymptotic expansion theorem. We shall give the proof in Section 8.2. In Chapter 4, we shall formulate the composition of the functionals of a Poisson random measure and Schwartz distributions. In Chapter 5, we shall define the asymptotic expansion, and give its basic properties. The asymptotic expansion theorem will be proven in Chapter 6. In Chapter 7, we shall prove the asymptotic expansion of the stochastic differential equation and also give a sufficient condition that satisfies the uniformly non-degenerate condition. From Chapter 2 to Chapter 7, and Section 8.2 will be published in

Hayashi, M., " Asymptotic expansions for functionals of

a Poisson random measure.", to appear in

Journal of Mathematics of Kyoto University.

5

Acknoledgements: I am most grateful to Y. Higuchi for kind encourage­ment and guidance. I would like to thank Y. Ishikawa for reading this manuscript and for his valuable advice, and to M. Okamoto for introduc­ing me to this subject.

6

Chapter 2

Notation

2.1 Probability space and hypothesis

We set E = [0,1] x (Rdn{oy). Let E be the Borel a-algebra on E, and 'x(du) a a-finite and infinite Radon measure without atoms defined on (E, E). We define

')'(u) = Ixl for u = (s,x) E [0,1] X (RdO n {Ole) = E.

Throughout this paper, we suppose that the measure space (E, E, ,x) satisfies the following condition: Condition 1

f-1). fE'"'?(U) A 1'x(du) < 00, where ,),(u) A 1 = min{')'(u), 1} ;

f-2). there exists a E (0,2) and Co > ° such that for each p E (0,1]

f(p):= r ')'2(U) 'x(du) 2: eol\ J A(p)

where A(p) = {u E E; ')'( u) ::; p}. We shall use the notation a, ')'( u), and f(p) in the whole paper.

By a point measure on (E, E), we mean a measure w which has the form w(B) = Lj OUj (B), where Uj E E, BEE and OUj is the Dirac point measure at Uj. Define N(w, B) = w(B) for a point measure wand for BEE. We denote by n the space of all of point measures on (E, E) such that w( {u}) ::; 1 for each U E E, and by:Fo the smallest a-algebra such that N(·, B) is mea­surable for each BEE. Then it is well-known that there is the probability

8

measure P on (O,.ro) such that, under P, {N(·,B);B E E} is a Poisson random measure with the intensity measure A:

PI). for BEE with A(B) < 00, N(B) is a Poisson random variable with mean A(B);

P2). N(B) :...- 00, if A(B) = 00;

P3). for Bl, ... ,Bk E E, random variables N(Bd, ... ,N(Bk) are indepen­dent, if B1, ... , Bk are pairwise disjoint.

We denote

N(B) = N(w, B), /N/(du) = N(du) + A(du), N(B) = N(B) - A(B).

We write .r for the P-completion of .ro.

2.2 Maps c:+, c:-, and operators D, D*

We shall introduce maps e-;;, et (u E E) from 0 to O. They are defined as follows

e;:w(A) = w(A n {u}C), e~w(A) = e;:w(A) + lA(u).

For a functional F(w), we write (F 0 e-;;)(w) for F(e-;;w), and (F 0 et)(w) for F(etw). Remark that these maps are defined for a random parameter w, and are not well-defined for a random variable; F = 0 P-a.s does not always mean that F 0 et = 0 P-a.s. However these maps are well-defined as a stochastic process parametrized by u E E; F = 0 P-a.s. implies that Foe; = 0 A x P-a.e .. One can check that for each w,

(2.1) e;:W = w A(du)-a.e., N(du)-a.e.,

(2.2)

where (h, ()2 E {+, -}. In this paper, we say that the process {Zu; u E E} is integrable, if

E[1e /Zu/A(du)] < 00.

9

and denote by Z the class of all of integrable processes. In Section 1 in [12], one can find the following property: (2.3)

E[j Zuoc~ A(du)] =E[j ZuN(du)], E[j ZuA(du)] =E[j Zuoc;; N(du)],

for Z E Z. Now, we introduce the operator D and its dual operator D*. For a func­

tional F, the operator D is defined by

DuF = Foc~ - F.

For an integrable process ZUl the operator D* ( 0 x E I-t 0 ) is defined by

(2.4)

The notion" dual" follows from:

Theorem 2.2.1. Let F be in L2(O) and Zu be in Z. Suppose that

Then, we have

(2.5) E[L (DuF)Zu A(du)] = E[F D* Z].

Remark. See Theorem 2 in [ll}.

The operators D and D* are frequently said to be the annihilation and the creation operator, respectively. These notions follow from analysis on the Fock space associated to the Hilbert space L2(E, 8, A) (see [10]). In particular, it follows from analysis on a Fock space that the operator D is closable.

For functionals F and G, we shall use the following rule of D:

(2.6) Du(FG) FDuG + GDuF + DuFDuG. (2.7) Foc~ F + DuF,

(2.8) F 0 c~l 0 c~2 - DUI DU2F + DUI F + DU2F + F.

10

By the definition of D and (2.2), we have also

(2.9) Du(F 0 c;;) = O.

Frequently, we shall consider the product measure space (Ek, &i£, 0 k,X), and use the convention EO = 0, ctw = w, and D0F = F. For a non-negative integer k, and for fJ = (UI, ... ,Uk) E Ek, we denote

c+ = c+ 0'" 0 c+ c- = c- 0'" 0 c-eF Ul Uk' eF Ul Uk DeFF = DUl ... DUkF.

If we need to express the length of fJ = (U1," ., Uk), then we denote D!F = DeFF. The formula (2.6) can be generalized as follows:

(2.10)

where CI'Td,I'T21 is a constant which depends on the length of 71 and 72. If F takes values in R d, then we define DeFF = (DeFF1 , ••• , DeFFd).

2.3 Spaces of random variables

Here, we shall introduce some spaces of random variables. We denote by LP(O) the V-space on (0, F, P) and by II . lip the V-norm. If a functional F = (F1 , •.. , Fd) values in Rd then we define

IIFllv = E[IFIV]~, LV(O;Rd) = {F;Rd-valued random variable and IIFllp < oo}.

For a non-negative integer k, we extend the function 'Y and the measure 'x(du) by putting

'Y(fJ) = 'Y( U1) ... 'Y(Uk) , 'x(dfJ) = 0 k ,X (dfJ) = ,X(dU1) ... 'x(dUk)'

In the case k = 0, we also use the convention 'Y(0) = 1. For a random variable F which takes values in R d, for p E (0,1), for p 2: 2, and for a non-negative integer k, we define

(2.11) IFIDk,p,p(Rd ) = [E[IFIP] + t ess ~up E [I D(tF) IV]] ; , j=l eFEA] (p) 'Y fJ

where the essential supremum is relative to the measure 'x(da).

11

Remark. This norm corresponds to the condition (a) in Theorem 2.1 of Picard (12j: for each p, k, IID~Fllp ~ Cp,k'Y(O") .:\(dO")-a.e ..

We shall use the convention

IFIDO,p,p(Rd) = IIFllp, IFlk,p,p = IFIDk,P,p(R1 ), IFIDk,P(Rd ) = IFIDk,P,l(Rd).

Definition 2.1. We denote by Dk,p(Rd) the set of all F E LP(fl; R d) for which there is a sequence {Fn : n = 1,2, ... } C LP(fl; R d) going to F in LP(fl; Rd

), such that IFnIDk,p(Rd) < 00 and IFn - FmIDk,p(Rd) -+ 0 as n, m -+ 00. We define DuF = limn--+oo DuFn. We denote Doo(Rd) = np>2 n~o Dk,p(Rd). In the case d = 1, we write Dk,p for Dk,p(R), and D~ for np~2 n~o Dk,p'

Remark. We shall show the closability of D with respect to the norm l'IDk,p' For this, we shall show the closability of D : L2(fl) t-+ L2(n x A(I)) in the first place. Let 'Pn be the set of permutations of {I, 2, ... ,n}. We define

(L2(E))0n := {f E L2(En); f(uu(l),'''' Uu(n») = f(uI, ... , un), "10" E 'Pn}.

Recall that L2(fl) can be considered as the Fock space 61:'=0 Cn, where Cn is the Wiener chaos of n-th order:

As we mentioned above, D can be considered as the annihilation operator: DuIn(J) = nIn- 1 (J(', u)). Hence, we have

E[ r IDuIn(J) 12 .:\(du)] ~ E[j IDu1n(J) 12 .:\(du)] 1 A(l)

= n2E[j IIn_1(J(·,u)12.:\(du)]

= nn! r If(O") 12 .:\(dO") = nn! E[lIn(J) 12]. lEn This means that the restriction of the operator D on each chaos Cn is closed. Hence, the operator D : L2(fl) = 61:'=0 Cn t-+ L2(fl x A(I)) is closed (see Section 8.1). Now, suppose that Fn -+ 0 (n -+ 00) in L2(fl) and that

II DuFn - Zu II esssup -+ 0

UEA(l) 'Y( u) 2 (n -+ 00)

12

for some process Zu. Then, we have

I

E[ { IDuFn - Zul2 A(du)]t :::; ({ 1'2 (U)A(dU)) "2 esssup II DuF() Zu II

} A(l) } A(l) uEA(l) l' U 2

-+ 0,

as n -+ 00. Since D : L2(n) I--t L2(n x A(1)) is closable, we have Zu = 0, P x A( du) -a.e.. This means D is also closable with respect to the norm 1·IDk,P. Therefore, the space Dk,p(Rd

) is a Banach space for p ~ 2.

One can check that, for k :::; k' and for p :::; p' .

(2.12)

and that Doo is an algebra, that is, F, G E Doo implies FG E Doo For a complex valued process Zu ( u E E), we define

IZIk,p,p := [t ess sup E [I F~Z() IP] 1 ~ , j=O (u,u)EA(p) l' (j l' u

where we used the convention IZIO,p,p = ess SUPUEA(p) E [I J:) n ~ . We define

00

Dk,p = {Zu; IZIk,p,p < oo}, D~ = n n Dk,p. p2:2 k=O

Lemma 2.3.1. Let PbP2, and r be positive numbers satisfying Pl,P2,r ~ 1 and 1.. + 1.. = ! < 1. For any non-negative integer k, suppose that PI P2 r-

IF(i) Ik,Pi,P < 00, Iz(i) Ik,Pi'P < 00 for i = 1,2. Then, there is a positive constant C = C(PbP2, r, k) such that the following inequalities hold

(2.13)

(2.14)

(2.15)

(2.16)

IFI F2Ik,r,p

IZ1 F21'" k,r,p

IZ1 Z21'" k,r,p

11 z~z~ A(du) I A(p) k,r,p

where r is given in Condition 1.

< CIFllk,PI,pIF2Ik,P2,P' < CIZ11'" IF21 k,pl,p k,P2,p, < C IZll'" IZ21'" p k,pl,p k,P2,P

< Cr(p) I zllk,PI,P IZ2 1k,P2,P'

13

Proof. We prove (2.15) only, because (2.13),(2.14) (2.16) can be proved in a similar way. Applying the formula (2.10), we have for a E Al(p) (l ::; k),

where the sum is given by (2.10). Because ,(u) ::; p on A(p), we have ,(rd,(-r2) ::; ,(a). Hence, we get

(Di Z(l»)(Dj Z(2») Di Z(l) Dj Z(2) esssup 71 U 72 U ::;p esssup 71 u 72 U

(u,U)EAI+1(p) ,(uh(a) (u,u)EAI+1(p) ,(rlh(u) ,(r2h(u) r r

Di Z(l) Dj Z(2) ::;p esssup 71 u esssup 'T2 u

(u,7!)EAi+1(p) ,(rd,(u) (U,72)EAi+1(p) ,(r2h(u) P1 P2

where, in the last inequality, we have used Holder's inequality with p~ r;-,p~ = ~. This proves (2.15). 0

Next, we denote Dk,p by the (analytical) adjoint space of Dk,p, that is, the space of all of linear continuous functionals on Dk,p. It is well-known that Dk,p is a Banach space with respect to the norm

1~1~,p:= sup I(~,G)I (~E D~,p). IGlk,p9

We define D~ := U U D~,p, 0:':= U n D~,p.

k=Op~2 k=Op>2

Composites of Schwartz distributions and functionals of a Poisson random measure will be defined as elements in 0:'. Note that we do not choose the space Uk=O np~2 Dk,p instead of 0:', see Remark 3.

14

Chapter 3

Preliminaries

3.1 Remarks on the domain of the operator D*

In this chapter, we shall exhibit preliminary results on functionals of a Pois­son random measure which play an important role in the whole paper. In particular, we shall give a modification of the smoothness criterion of Picard. To this end, we shall study the domain of D*.

The positive constant numbers will be denoted by C and may vary from line to line. If they depend on some parameters, then this is emphasized by index.

We introduced norms II . IIDk,P and II . IIDIc,p in the previous chapter. To study the domain of D*, we shall introduce other norms

(3.1)

where AO (du) is the probability measure

(3.2) 1A(1) (uh(u)2

Ao(du) = JA(l) 1'2(U)A(du) A(du).

16

Then, the following inequalities are true

(3.3) IZI'" < IZI'" Wk,p - Dk,p'

for a functional F E Dk,p and a process Zu E Dk,p. Let S be the collection of random variable X written as

where f(xI, ... , xn) is a bounded smooth function in XI, ... Xn in R d , n E N, and CPl,' .. , CPn are smooth functions on E with compact support. Note that IXlk,p < 00. From (3.3), we have IXlwk,p < 00 if XES. Let Wk,p be the completion of S by II . IIwk,p' We denote Woo = n:o np>2 W k,p, and W~ = n:o np>2 Wk,p' In [5], these norms (3.1), and spaces Wk,p and W'k,p were introduced-and discussed.

Theorem 3.1.1. Let Zu be a stochastic process belonging to Z n W~, and p an even number, and k a non-negative integer. Then, there is a constant C = C(p, k) > 0 such that

(3.4)

holds.

Remark. Ishikawa-Kunita have shown slightly stronger estimation than (3.4) for Wiener-Poisson functional. For the proof, see Theorem 3.2 in Ishikawa­Kunita {5}.

Note that the dual operator

is closable.

Lemma 3.1.1. Suppose that IZlw < 00 for any k, p, and that Zu = 0 k,p on A(l)c. Then, Z is in the domain of D*. Further, for any non-negative integer k and even number p, there is a constant C = C (p, k) > 0 such that

(3.5)

holds.

17

Remark. Recall that the operator D* is defined for elements in Z by (2.4). In Lemma 3.1.1, we said that the domain of D* is that of the dual operator of D. Hence, (2.4) does not necessarily hold for the process Z which satisfies the condition of Lemma 3.1.1. However, it can be deduced from the proof of Lemma 3.1.1 that

D*(Z) = lim r Zuoc;;'N(du), in Wk,p' 1-+00 i A(l)nA( t)C

From this, one can deduce that (2.5) holds for Z satisfying the condition of Lemma 3.1.1 and for F E L2(O) satisfying E[fE IDuFI2 A(du)] < 00.

Proof. For a natural number 1, we set Z~ = ZU1A(1)nA(t)c(U). Note that A( A( t)) < 00. Hence it holds that the process Z~ is integrable. Indeed, applying Schwarz's inequality, we have

Elf IZ~IA(du)1 " r( ~)' Elf 1 'Y1:) I' Ao(du)]',

where c = fA(l) ,2 (u)A(du). Note that Ao(A(t)) = r(t) ~ 0 as 1 ~ 00. It

also holds that Zl ~ Z in W'k,p for any non-negative integer k and p ~ 2, since Schwartz's inequality yields that

E[ r r 1 Du(Z~ - Zu) I

P

AO (dO') AO (du)] iA(l)iA(l)k ,(0')

= E[ r r IDu(Z)uIP

Ao(dO')Ao(du)]:::; r(~1)~IZlwk2 ~ 0, iA(t) iA(l)k , 0' ,p

as 1 ~ 00. Applying Theorem 3.1.1, we get

(3.6) ID*(Zh) - D*(ZI2)lw k,p :::; Ck,plZh - ZI2Iwk+P,k+P ~ 0

as h, h ~ 00, for any non-negative integer k and even number p. In particular IID*(Zh) - D*(ZI2) 112 ~ O. As we noted above, D* is a closable operator. This shows that Z is in the domain of D*. Because W k,p is a Banach space, one can can check that D*(Z) is in Wk,p for any k,p. We have

ID*(Z)lwk,P:::; ID*(Z) - D*(ZI)lwk,p + ID*(ZI)lwk,P

:::; ID*(Z) - D* (Zl) IWk,p + CIZ1A(l)nA(t)clw k+p,k+P'

Letting 1 tends to infinity, we get the inequality (3.5). o

18

Remark. From this lemma, we can also define D* Z if Z is in Dk,p for any k,p.

Pick a system Z = {Z}P;j = 1,2, ... } c D~ which satisfies z~j) = 0 on u E AC(l) for any j. For any G E Doo , it follows from Lemma 2.3.1 that GZ(1) is in D~. Hence, Lemma 3.1.1 shows that D*(ZG) can be defined and is in Woo. By the iteration of this argument, and by Lemma 2.1 in [5], one can define

(3.7) D;(n)(G):= D*(z(n)D*(z(n-1) ... D*(Z(1)G) .. . )),

and show that D~(n) (G) E Woo. Repeating the argument of the proof of Lemma 3.1.1, we have:

Lemma 3.1.2. Z = {ZIP;j = 1,2, ... } c W; which satisfy Z~j) = 0 on u E AC(l) for any j. Put zf!),l = zf!)lAC(i)(u) for a natural number l. Then, for G E Woo, we have

lim ID*(n) (G) - D*(n) (G) I -t 0 l-too Zl Z Wk,p ,

where D;~n) (G) is defined by (3.7) with z~j) = Z~j),l.

3.2 Smoothness criterion of Picard

The following theorem is a modification of Theorem 2.1 of Picard [12]. Re­call that Condition 1 in Section 2.1 holds, and that we use the notation O!, ,),(u) , r(p) in the whole paper.

Theorem 3.2.2. Suppose that FE Doo(Rd) satisfies the following condition: (ND). there exists some (3 E (%,1] such that for any p E (1,00), any p E (0,1), and any non-negative integer k,

sup ess sup II (r Iv· DuFI21{1v.DuFI::;pf3}>'(du) ) -loci II S; Cp,kr(p)-l, VERd TEAk(p) } A(p) P Ivl=l

where Cp,k does not depend on p. Then, for any natural number n, for r > 2, for G E Doo, and for e E Rd, we have

(3.8) sup /E[Geie·FJI S; C(l + I e 12)-(1-2P)~ IGln.r=l

where C is a constant which depends on n, rand F.

19

Remark 1. One can weaken the non-degenerate condition (ND) to the fol­lowing form;

sup esssup II (r lev.DuF - 112 'x(du) ) -1 Oc; II ~ Cp,kr(p)-I,

VERd TEAk(p) J A(p) P Ivl=l

because, in [12J, the author used the non-degenerate condition (ND) to esti­

mate the random variable ( IA(p) lev .DuF - 112 ,x (du) ) -1 oct. See Section 2

in [12].

Remark 2. Suppose that F = (F1, ••• , Fd ) E Doo(Rd) satisfies the condition (ND). We set 1/J~(e) := E[GeiEF]. Because Doo is an algebra, .Ffl ... F:;"d is also in Doo for each multi-index m = (m1, ... , md). From the inequality (3.8), we obtain

I :e;l ... :e;d 1/J~(e) 1=1 E[GFfl ... F:;"dei{F] I ~ C(l + leI2)-(1-~)~,

where C is a constant which depends on n, m, F, and G. This means that 1/J~ is a rapidly decreasing function, hence so is the Fourier inversion of 1/J~. In particular, F has the density function which is rapidly decreasing. We denote by PF(X) the rapidly decreasing density function of F. Then, because one can write 'I/J<j.(e) = I ei{XE[GIF = X]PF(X) dx, the function :F1/J<j. := E[GIF = X]PF(X) is also rapidly decreasing.

Remark 3. It seems difficult to obtain the inequality (3.8) for r = 2. We shall prove Theorem 4.0.1 by using (3.8) and define the composites of Schwartz distributions and functionals of a Poisson random measure. This is the rea­son why we define :5:0 as in Section 2.3.

Theorem 3.2.3. Let Z~j) (e) ; j = 1, 2, ... be processes parametrized by (u, e) E E x Rd. Suppose that there exists some ~ < (3 ~ 1 such that Z!P(e) = 0 if

1

U E A(I e I-:a), and that, for any non-negative integer j, k, for p ~ 2 and for any e E {e E R d; 1 e 1 ~ I},

IZ(j) Ik,p,IE r! ~ Ck,p,j( 1 e Ir(1 e I-~) )-1

where Ck,p,j does not depend on e. Then for each G E Doo and r > 2, it holds that

sup IID~~~(G)lb ~ C(l + 1 e 12)-(1-2P)~, IGln,r=1

20

where D~~~~(G) is defined by (3.7) with zfP = ZfP(e). Remark. The proof is essentially due to Picard, although slight modifications are needed. We shall prove this theorem in Section 8.2.

Here, applying Theorem 3.2.3, we prove Theorem 3.2.2. Put

(3.9) (e-i~.DuF - 1)1 _1 (u)

( A(I~ I If)

Zu e) = f _1 lei~'DlIF -112)'(dV)' A(I~I If)

We shall use the following Lemma of Picard (see proof of Lemma 2.8 in [12]);

Lemma 3.2.3. Let F be in Doo(Rd). Then it holds that, for any non­negative integer k for p ~ 2 and for any e E {e E Rd; 1 e 1 ~ I},

(3.10)

and that

(3.11) IZ(·, e)lk,p ~ Ck,p,j (I e 1 r _1 'l(U),(du))-1 ) ACI ~ I If)

Remark. Lemma 2.8 in [12} claims only that the inequality (3.11) holds, but in the proof, it is shown that (3.10) also holds. We shall use the inequality (3.10) in Chapter 6.

Note that

hence, we have

ei~.F = / Zu(e)Duei~F )'(du).

It follows from Lemma 3.1.1 that

Repeating this argument, we get

21

The absolute value of the right hand side is dominated by

IID*(Z(e)D*(Z(e)··· D*(Z(e)G) .. ·))112,

Therefore, applying Theorem 3.2.3 with Z~.P = Zu(e), we complete the proof of Theorem 3.2.2.

Remark. As we mentioned above, the proof is essentially due to Picard. However, the process Z(e) is differ from that of Picard {12}; instead of Z(e), the author used the following integrable process

I _1 leie·DvF-112).(dv)' A(I e I 7f )nA( ()c

where ( > 0 small enough. The process Z(e) defined by (3.9) is not neces­sarily integrable. However, to prove asymptotic expansion theorem, we have to show Theorem 3.2.3 for z(j) 's which are not necessarily integrable.

22

Chapter 4

Composites of functionals of a Poisson random measure and Schwarz distributions

In this chapter, we shall formulate composites of functionals of a Poisson random measure and Schwartz distributions. Our formulation slightly differs from that of the case for Wiener functionals (see Watanabe [17]). Take a natural number d and be fixed. We shall consider Schwartz distributions on Rd. We denote by S the space of rapidly decreasing functions on Rd. Recall that S is a Frechet space. We denote by S' the space of continuous linear functionals on S, that is the space of tempered distributions. For <I> E S, we write :F<I>(~.), P<I>(x) for its Fourier transform and its inverse Fourier transform, respectively:

For <I> E Sand s E R, we define

and denote by Hs the completion of S by II . IIHa' We also denote Hoc = ns>oHs, and H-oo = Us>oH-s. Then it is obvious that for s < r

(4.1) 1I<I>IIHa :S 1I<I>IIHr' S C Hr C Hs C S'.

24

For a functional F, we define AF¢ = ¢(F) (¢ E S). Since for any k, the map Dk,p 3 G H E[GAF¢] is a linear continuous functional on Dk,p, we can regard AF as the operator

(4.2)

Theorem 4.0.1. For F E Doo(Rd), we suppose that F satisfies the condition

(ND) in Theorem 3.2.2. Then for any s > 0, there exists a natural number n such that

(4.3) IAF(¢)ID:,p ~ CII¢IIH_., for all ¢ E S, and any p > 2,

where C depends on F, n, p and s.

Remark. This theorem corr:esponds to Theorem 1.12 in Watanabe f1J} for Wiener functionals.

1 v( 2. )

Proof. For 8 > 0, take an integer n so that n ~ 1-1:' (1 + d). Then, one 2/3

can check that fRd(l + leI2)-(1-2p)~ de < 00. Note that

¢(F) = (J::F¢)(F) = r eiF·e:F¢(e) de. iRd

By Theorem 3.2.2 and using Schwarz's inequality, we have

IAF(¢)ID* = sup IE[G¢(F)] I n,p

IGln,p=1

= IG~~;=1 IE [ fad GeiF·e:F¢(e) de] I ~ r sup I E[GeiF·e] I 1:F¢(e) I de iRd IGln,p=l

~ C r (1 + leI2)-(1-2p)~I:F¢(e)1 de iRd ~ C [ fad (1 + leI2)-(1-2P)~ de ] ~ II¢IIH-(l-~)~ = C'II¢IIH_(l-271)~ ~ c'II¢IIH_.,

where, in the last inequality, we used the fact that -(1 - ~)~ ~ -8 and (4.1). 0

25

Because S is dense in H s , the inequality (4.3) shows that the linear op­erator (4.2) has the unique continuous extension:

...... * (4.4) AF : Hs :3 T r-t AFT E Doo.

From this fact, we define composites of Schwartz distributions and functionals of a Poisson random measure:

Definition 4.1. Suppose that F = (F1 , .•• , Fd ) satisfies the condition (ND) in Theorem 3.2.2. For any T E H-oo, we say that the linear continuous functional AFT is composite ofT E H-oo and F, and denote To F = AF.

In the Malliavin calculus on Wiener space, composites of smooth Wiener functionals and Schwartz distributions can be evaluated by integration by parts formula ( see Watanabe [17]). On the other hand, as we mentioned in Chapter 1, we cannot apply the integration by parts formula in our formulation. However, from Theorem 3.2.2, we know that the function 'IjJ~(~.) = E[Geie·F] is in S. Hence, the inequality (4.3) yields

To F : G r-t (T 0 F , G) = s' (:FT , 'IjJ~)s.

Note that :F'IjJ~(x) = E[GIF = X]PF(X), where PF is the rapidly decreasing density function of F. Hence, the following equality also holds

(4.5) ( T 0 F , G) = s' (T , :F 'IjJ~ ) s.

Next, we define the product H E Doo and To F by

(HTo F, G) := s,(:FT, 'IjJ~,H)S (G E Doo),

where 'IjJ~,H (~.) = E[GHeie·F]. From Theorem 3.2.2, 'IjJ~,H is in S. By the same argument as in the proof of Theorem 4.0.1, one can deduce that HT 0

...... * FE Doo.

Example 4.1. If T is the Dirac point mass Ox at x, then it follows from (4.5) that

(T 0 F, G) = E[GIF = X]PF{X).

If T is the Heavisde function 1[0,00), then we have also

(ToF,G) = E[G;F ~ 0].

26

Chapter 5

Asymptotic expansions

5.1 Asymptotic expansions

In this chapter, we shall consider a family of functionals F(€) (€ E (0,1)) depending on the parameter €.

Definition 5.1. A) We say that F(€) has the asymptotic expansion F(€)rv Ei=o €j!; in Doo(Rd), if the following conditions hold:

A1) F(€), fo, /I, ... , E Doo(Rd) for each € E (0,1).

A 2) For each non-negative integer m, k and for p ~ 2,

B). We say that ~(€) E :5~ has the asymptotic expansion ~(€) rv Ei=o €j~j in :5:0, if the following conditions hold:

B1) ~(€), ~o, ~b"" E :5~ for each € E (0,1).

B2) For each non-negative integer m, there exists a k = k(m) such that ~(€),~O'~b""~m E np>2Dk,p and

I ~(€) - ~'?'" €jq> ·1* I· L..JJ=O J k,p < 1m sup m+l 00.

f--+O €

Remark. In condition B2), we restricted p to be in (2,00). This restriction arise from the definition of :5:0.

28

We set ud := {e E R d : I e I ~ 1}.

We shall consider complex valued random variables has the form F(E, e) where (E, e) E (0,1] X Ud

, and give another definition of asymptotic expan­sions. Let q(e) be a positive functions defined on Ud

, pee) be a function defined on Ud and taking values in (0,1]. Let F(E,e) (E,e) E (0,1] X Ud be a family of elements in Doc. For a non-negative integer m, we denote

if it holds that, for any p ~ 2 and for any non-negative integer k,

IF( E, e) Ik p p(~) lim sup sup " .. < 00. £-+0 tEUd Emq(e)

(5.1)

Definition 5.2. A'). We say that a complex valued random variable F( E, e) has the asymptotic expansion F(E, e) rv 2:;:0 Ej lice) in Doc(q(e), A(p(e))), if the following conditions hold:

A'l) for any (E,e) E (0,1) x Ud, F(E,e),fo(e),!I(e) ... ,E Doc.

A' 2) For any non-negative integer m

m

F(E, e) - L Ej fj(e) rv O(Em+1q(e)) on A(p(e))·

j=O

A'3) For any non-negative integer j, k, for any p ~ 2 and for anye E Ud ,

1/i(e)lk,p,p(t) ~ ej,p,kq(e)·

e'). We say that a complex valued process Zu(E, e) has the asymptotic ex­pansion Zu( E, e) rv 2:;:0 Ej z~)(e) in D~(q(e), A(p(e))), if the following conditions hold:

e'l) Zu(E,e),ziO)(e),z!P(e), ... ,E D~, for any (E,e) E (0,1) x Ud•

e'2) For any non-negative integer m,

m

Z(€,e) - L€jzU)(e) rv O(Em+1q(e)) on A(p(e)). j=O

29

C'3) For any non-negative integer j, k, for any p 2:: 2 and for any e E

ud , Iz(j)(e)lk,p,p(() :::; Cp,kq(e).

C'4) For any j, Zu(€,e) = z~)(e) = 0 ifu E AC(p(e)).

Remark 4. One can check that the coefficient of the asymptotic expansion is uniquely determined.

By using Lemma 2.3.1, one can easily prove the following proposition.

Proposition 5.1.1. Let qi(e), q;(e) (i = 1,2) be non-negative functions de­fined on Ud, and p(e) be a function defined on Ud which values in (0,1]. Suppose that for i = 1, 2

00

Fi (€, e)"-' L €j f](e) in Doo( qi(e), A(p(e)) ) (i = 1,2),

00

Z!(€,e) "-' L€jz~),i(e) in D;(q~(e),A(p(e))) (i = 1,2). j=O

Then, it holds that

i). 00

F1(€,e)F2(€,e) "-' L€jhj(e) in D;(ql(e)q2(e),A(p(e))), j=O

ii). 00

Zl(€,e)F1(€,e) "-' L€jx(j) in D;(ql(e)q~(e),A(p(e))), j=O

iii).

00

Zl(€, e)Z2(€, e) "-' L €j z(j)(e) in D;( p(e)q~ (e)q;(e), A(p(e)) ) j=O

30

iv).

00 J Z~ «, e)~ «, e)A( du) ~ f;,J 9; (e) , in Doo( r(p( e) )q; (e)q; (e), A (p(e)) ),

5.2 Asymptotic expansion theorem

For a multi-index n = (nb"" nd) and for x = (Xl,"" Xd) E Rd, we denote

n! = nl! ... nd!, xn = X~l •• 'X~d.

Definition 5.3. Let F(€) (€ E (0,1]) be a parametrized process such that F(€) E Doo(Rd) for any € E (0,1]. We say that F(€) satisfies the uniformly non-degenerate condition if

(UN). there exists some ~ < f3 ~ 1 such that for any p E (1,00), and any non-negative integer k,

limsup sup supesssup/lr(p)(K(€,P,v)O€;)-l/l <00, €-+o PE(O,l) vERd rEA"(p) P

Ivl=l

The following theorem is our main result:

Theorem 5.2.1. Suppose that F(€) (€ E (0,1]) satisfies the uniformly non­degenerate condition (UN) and that F(€)rv 2:;0 €j!; in Doo(Rd). Then, for

any distribution T E H-oo, To F(€) E fi~ has the asymptotic expansion in fi~(Rd):

00 1 To F(€) rv L L n! (anT) 0 fo . (F(€) - fot

m=Olnl=m

rv 4> + €4>1 + €24>2 + ... ,

31

where cI>, cI>I, cI>2, ... , are given by the formal Taylor expansion

d . 8 cI>0 = To fo, cI>1 = ~ n(8xi T) 0 fo,

i=l

d. 8 1d .. 82

cI>2 = ~f~(8xiT) 0 fo + 2! ij; fUi (8xi8xj T) 0 fo

d. 8 2d .. 82

cI>3 = ~ f~(8xiT) 0 fo + 2! ~ fU~(8xi8xjT) 0 fo i=l i,j=l

Remark. We shall give the proof of Theorem 5.2.1 in Chapter 6.

The following theorem is a sufficient condition to satisfy the uniformly non-degenerate condition (UN), and is a version of Theorem 3.1 of Picard [12].

Theorem 5.2.2. Suppose that >"(ds x dx) = ds x v(dx). Define the d x d matrix V(p) by Vi,j(p) = f{lxl::;p} xixjv(dx). Suppose also that the ratio between the largest and smallest eigenvalues of V(p) is bounded as p ---t 0, and that

liminf p-o r IxI 2v(dx) > O. p-+O J{jxl::;p}

The random variable F(i) E Doo satisfies the uniformly non-degenerate con­dition if the following conditions (a) and (b) hold: (a). for any p > 1 and for any non-negative integer k

sup esssup I D~r~i) I < 00, f TEAk(l) 'Y T

P

(b). there exists a matrix-valued process 'ljJt(i) such that for Ixl ~ 1 and for any p ~ 1,

sup IIDt,xF(i) - 'ljJt(i)xll p ~ Cplxl r,

f

32

for some r > 1, and

(5.2)

where 'IjJ;(€) is the transpose of the matrix 'ljJt(€).

Remark. Note that the conditions (a) and (b) are uniform in €. Therefore, Theorem 5.2.2 can be proven by a similar argument as Theorem 3.1 in [12J, so we omit the proof.

33

Chapter 6

Proof of Theorem 5.2.1

Let f3 E (I' 1) be fixed. We define functions p(e), ql(e), q2(e) on Ud by

1

p(e) = lel-p , ql(e) = lei, q2(e) = leI2r(p(e)).

Lemma 6.0.1. Suppose that G(E, e) rv E~o Ei gi(e) in Doo(Q2(e), A(p(e))), and that for any p 2 2 for any non-negative integer k, and for e E Ud ,

(6.1) lim sup sup esssup IIQ2(e)(G(E,e) 0 c~)-lilp < 00. €-tO eEUd TEAk(p(e))

Then, G-l (E, e) has the asymptotic expansion

00

(6.2) G-1(E,e) rv LEi g~(e) in Doo(Q21(e),A(p(e))), i=O

where the coefficients are given by the formal expansion:

Remark. In the proof below, we shall show that

(6.3)

34

The argument of the proof of iii} in Lemma 2.7 of Picard [12J show that (6.1), and (6.3) yield (6.4)

lim sup sup Iq2(e)G-1(f,e)lk,p,p(E) < 00, Igo1(e)lk,p,p(E)::; Ck,pq21(e),

£-to EEUd

respectively.

Proof. Firstly, we shall prove (6.3). For any r > 0, applying Chebychev's and Schwarz's inequality, we get

esssup p (Igo(e) oct 1< r) TEAk (p(e)) q2(e)-

::; esssup P (I G(f, e) oct I ::; r + I G(f, e) - go(e) I oc;) TEAk (p(E)) q2(e) q2(e)

::;Cp

esssup l(r+IG(f,e)-go(e)1 oc;)I·IG(f,e)oc;I-1

P TEAk (p(E)) q2(e) q2(e)

p

< C (rP + esssup II (G(f,e) - go(e)) oc+IIP ) esssup II q2(e) liP - p TEAk (p(E)) q2(e) T 2p TEAk(p(E)) IG( f, e) 0 ctl 2p By iteration of (2.8) and by the assumption, we have

lim sup esssup II (G(f,e) - go(e)) oc;IIP = 0

t"-tO EEUd TEAk(p(E)) q2(e) 2p Hence, from (6.1), we have

sup esssup P (Igo(e~ o)c; I::; r) ::; Cp,k rP . EEUd TEAk (p(E)) q2 e

This implies (6.3). We shall show (6.2). On the event B := {I G(£,:~~)o(E) I < 1 }, we have

1 __ 1_ 1 _ 1 ~ j (G(f,e) - go(e))i G(f,e) - go(e) 1 + G(£,E)-go(E) - go(e) ~(-1) 9 (e) . ~~ J~ 0

Hence, we can write

35

By the assumption, we see that, for any p ~ 2 and k, there exists EO > 0 such that IG(E, e) - gO(e)lk,p,p(e) ~ Cp,kEQ2(e) holds for any E E (0, Eo). Hence, it follows from (2.10) and (6.4) that

< C Em+1Q (1:)-1 _ p,k 2 I!O ,

for E > 0 small enough. On the other hand, applying Chebychev's inequality, we have

(6.6) P(BC)$ < C IIG(E,e) -go(e)llm

+1 < C Em +1 - p,m (I:) - p,m , go I!O 2p(m+1)

for E > 0 small enough. One can check that

< esssup - ; B C

II Du 1 II

- uEAk(p(e)) ,(0") G(E, e) p

+ t esssup Du (_l_(_l)i (G(E,e) - go(e))i) . BC

i=O UEAk(p(e)) ,(0") go(e) go(e) 'p

It follows from Schwarz's inequality, (6.4) and (6.6) that the right hand side is bounded by Cp,kEm+1Q2(e)-I. Combining (6.5), we have

1 1 ~ i (G(E, e) - go(e))i Em+1

G(E, e) - go(e) ~(-1) go (e) ~ Cm,k Q2(e) , - k,p,p(e)

for E > 0 small enough. Further, Proposition 5.1.1 yields that, for any j,

This completes the proof. o

36

Lemma 6.0.2. Suppose that F(€) rv 'L::=o€nfn in Doo(Rd). We set

Xu(E,e) = (eie·DuF(€,e) -l)lA(p(e))(u).

Then, we have Xu(€, x) rv 'L';o Ejx~)(e) in D;(ql(e), A(p(e)))· In particu­

lar, x~O)(e) is given by (eie·Dufo(e) - l)lA(p(e)) (u).

Proof Put x~O)(e) = (eie·Dufo(e) - l)lA(p(e)) (u). Then, one can write

Xu(E,e) - x~O)(e) = (eie·Du(F(€)-fo) _l)eie·Dufo(e)lA(p(e))(u).

It follows from (3.10) that Ix(O)lk,p,p(e) ~ Cp,kql(e) and leie·D.fo(e)lk,p,p(e) ~ Cp,kQl(e). Hence, if we can show that (eie·D,,(F(€)-fo) - l)lA(p(e))(u) has the

asymptotic expansion 'L';l Ejh~) in D~(Ql(e),A(p(e))), then the assertion

follows from (2.15) and the fact that p(e)(Ql(e))2 = I e 12-~ ~ Ql(e)· We set Hu = Du(F(E) - fo). It follows from the proof of Lemma 3.2.3

that

(6.7)

for E > 0 small enough. Note that

p

II (e 'Hu(€))m (e-i~.Hu(€) - 1) II

< sup ess sup Du . - JLE(O,l) (u,u)EA(p(e)) ')'(ah(u) p

As we mentioned above, p(e)(Ql(e))2 ~ Ql(e) holds. Hence, it follows from (2.15) that the right hand side is bounded by €m+1 Q1 (e) for E > 0 small enough. From Proposition 5.l.1, we know that (e 'Hu(€))j has the asymptotic expansion 'L::=j enh~(n) in D~(QI (e); A(p(e))). Therefore, eie 'Du(F(€)-fo) -1

has the asymptotic expansion 'Lj:l Ej h~) in D;(Ql(e), A(p(e))). 0

37

Lemma 6.0.3. Suppose that F(E) rv 2:;'oEj Ii in A(l), and F(E) satisfies the uniformly non-degenerate condition. Put

(6.9)

Then, we have

In particular, we have

(6.10) esssup TEAk(p(E))

Proof. Note that

p

IG«,~) 0 £;1 ~ Ie 12 (Lcp(e» II ~ I . DvF(,f l{ID.FC,)I~le 1-') (u)A(du) ) 0 £;.

Hence, uniformly non-degenerate condition yields that

lim sup sup esssup Ilq2(e) (G(E,e) O€;)-lll < 00. E-tO+ EEUd TEA(p(E)) P

Because, G(E, e) = JA(p(E))(eiE'DuF(E) _l)(e-iE·DuF(E) -l)'\(du), the assertion follows from Lemma 6.0.2, iv) in Proposition 5.1.1, and Lemma 6.0.1. 0

The following theorem is easily deduced from Lemma 6.0.3, Lemma 6.0.2 and ii) in Proposition 5.1.1.

Theorem 6.0.1. Suppose that F(E) rv 2:;'0 Ej Ii in A(l), and F(E) satisfies the uniformly non-degenerate condition.

We set

(6.11) _ (e-iE·DuF(E) - l)lA(p(E))(u)

Zu(E,e) - I leiE·DvF(E) -112'\(dv)' A(p(E)

Then, we have Z(E, u, e) rv 2:;'0 Ej Zj(u, e) in D~(qo(e); A(p(e))), where

38

Lemma 6.0.4. Suppose that F(€) ~ E;o €i Ii, and F(€) satisfies the uni­formly non-degenerate condition. We set

Then, for any natural number n for any r > 2 and for e E ud, we have

(6.12) lim sup sup sup 1€-(m+1)(1 + I e 12)-(m+1)+(1-~)~ E[GRm(e, €)]I < 00.

€-+o eEUd IGln.r=l

Proof. It follows from Proposition 5.1.1 that (~. (F(€) - fo))i has the asymp­totic expansion E~i €kh~(e) in D(I eli; A(p(e))). From Lemma 6.0.3, the inequality (6.10) holds. Hence, from Theorem 3.2.2 (see Remark 1), we have

for € > 0 small enough. We can regard G ~ E[Gei!/o €i hj(e)] as a linear continuous functional on Dn,r' Therefore, there is a sequence {In(e,·) ; n = 0, 1, 2, ... } C D~,r such that, for € > 0 small enough,

m ie.fo m

sup :EE[G;'(~. (F(€) - fo))k] - :E €klk(e, G) IGln.r=l k=O' k=O (6.13)

~ c€m+ll e Im+1-(l-~)~. By the Taylor expansion, we have

m ie.fo (6.14) eie ·F (€) = :E Ti(e ·(F(€) - fo))k + O((€I e IIF(€) - fol)m+l).

k=O

Hence, it holds that for each m

m

(6.15) sup E[Geie ·F (€)] - :E €klk(e, G) ~ c€m+11 e Im+l, IGln.r=l k=O

for € > 0 small enough.

39

On the other hand, we define Zu (€, e) by (6.11). Then, as we saw in Chapter 3, we have

E [Gei~.F(f)] = E [D;~~~~)(G)ei~'F(f)] ,

where D;~~~~)(G) is defined by (3.7) with Z~P = z~j)(€,e). It follows from

Theorem 6.0.1 that Zu(€,e) has the asymptotic expansion L;:o€j z~)(e) in D;(qo(e), A(p(e)))· Hence, we have

(6.16) m

E [D~~:~J)(G)ei~'F(f)] = E [D* (( Z(€,e) - I>jz(j)(~)) D~\:~~)(G) ) ei~'F(f)] j=O

m

+ L€jE [D:;({) (D~~:~~)(G)) ei~'F(f)]. j=O

From Theorem 3.2.3, we have

IG~~;=l E [D' (( Z«,e) - t,<;Z;(~)) D~\~~;>(G)) e;{OF(')]

S; cn €m+11 e 1-(1- 2P)~ •

Similarly, by expanding the sum L7=o€jE [D:j(~) (D;~~)I)(G)) ei{.F(f)] as

in (6.16), we get

m

sup E[Gei~.F(f)] - L€jl~(€,e,G) S;Cm,n€m+1lel-(I-~)~, IGln,r=1 j=O

where l~ (€, e, G) is a linear continuous functional on Dn,r and is given by

l~(€, e, G) = L E [D;h(~) (D;i2(~)(' .. (D:;n(~)(G) ... ))eie.F(f)] . jl+'+jn=j

Again, from Theorem 3.2.3, we have sUPIGln,r=1 Il~( €, e, G) 1 S; CI e 1-(1- ~)!i. Therefore, by the Taylor expansion (6.14), we can pick the sequence {lk' (e, .); k = 0,1,2, ... } of linear functional on Dn,r such that, for each m,

m

(6.17) sup E[Gei~.F(f)] - Llk'(e, G) S; Cn ,m€m+1le Im+1-(1-2p)~, IGln,r=1 k=O

40

for € > 0 small enough. If we compare (6.15) with (6.17), then we get lk(e,·) = lk'(e,·) for any k. Therefore, from (6.13) and (6.17), we get

m

sup IE[GRm(e,€)]1 ~ sup E[GeiE·F(E)] - L1k'(e,G) IGln.r=l IGln.r=l k=O

m iE.fo m

+ sup L E[GT(~' (F(€) - fo))k] - L lk(e, G) IGln.r=l k=O' k=O

~ cm€m+11 e Im+1-(1-2p)~, for € > 0 small enough. o Proof of Theorem 5.2.1. By the definition of the composition, we have

( L (-\OnT) 0 fo (F(€) - fo)n, G) n.

Inl:=;m

= L -\s' ( FonT, E[G(F(€) - fO)neiE'fo ] ) Se n.

Inl:=;m

= S' (FT, L (-i~)n E[G(F(€) - fo)neiE' fo ] ) S(' n.

Inl:=;m

Taking So > 0 so that T E H-SO and n so that n ;::: d-W~:tl. Then, we

have J(1 + I e 12to+m+1-(1-2p)~ de < 00. We define Rm(€, /) by (6.12). We get

II To F(€) - L ~(onT) 0 fo (F(€) - fo)n 1/* Inl:=;m n n,r

= sup I (To F(€) , G) - ( L -\(onT) 0 fo (F(€) - fo)n, G) I IGln.r=l Inl:=;m n.

= sup I s,(FT, E[GRm(e,€)])s( I IGln.r=l

~ IG~~;=l I/TI/H- so [Ld (1 + le12)SO IE [ GRm(e, €) ] 12 de ] ~

~ I/TI/H- so [1 (1 + lel2t o sup IE [GRm(e, €)W de ] ~ . Rd IGln.r=l

The assertion follows from Lemma 6.0.4. o

41

Chapter 7

Applications to stochastic differential equations

In this chapter, we suppose that E = (0,1] x Rand A(du) = ds x l/(du). As we mentioned in Chapter 1, we shall consider the following real valued stochastic differential equation:

(7.1) Xt(t) = xo + 1t b(Xs_(t))ds + t 1:! a(Xs_(t), y) N(ds x dy)

where € E (0,1), and give the asymptotic expansion: 00

(7.2) X 1(t) rv L tn In in Doo· n=O

For a function h(x, y) on R2, we denote by h(n)(x, y) the n-times derivatives with respect to x, if it is differentiable. The regularity and boundedness assumptions for functions a and b are the following:

Assumption 1. Suppose that there exist functions a(x) and a(x, y) such that

(7.3) a(x, y) = a(x)y + a(x, y).

The functions a(x) and b(x) are infinitely differentiable with bounded deriva­tive. The function a(x, y) is also infinitely differentiable with respect to x and satisfy the following conditions: for some positive constant TO > 1 such that f{lxl<l} Ixlrol/(dx) < 00, the following inequalities

(7.4) la(x,y)l:S: 0(1 + Ixl)lylro, la(n)(x,y)l:S: 0lylro on {y; Iyl :s: I},

42

hold. Further, suppose that the following inequalities hold, for any p ~ 2 and for n ~ 1,

(7.5) J la(x, y)IPv(dy) :S C(l + IxI)P, s~p J la(n) (x, y)IPv(dy) < 00.

Remark. Under Assumption 1, the stochastic differential equation (7.1) has the solution. Moreover, one can find several properties of the solution to the stochastic differential equation (7.1) which Picard {12} has obtained. For example, one can deduce from the proof of Theorem 4.1 in {12} that, for any k,p,

sup E less sup I D;~t if) IP

] < 00, f uEAk(l) 'Y a

in particular Xt(f) E Dco. Using the argument of the proof of Theorem 4.1, we can also show that

(7.6) sup sup IXt(f)lk,p < 00. f O<t~l

For convenience, we extend the region of the parameter f to the open interval (-1,1). We denote by :Fs the least a-field which N(A) (A C (0, s] x R}) are measurable.

Lemma 7.0.1. Let 0 :S So < 1 be fixed, and Yso,t(f) (0 < So < t :S 1) be a semimartingale having the following form

YsoAf) = Yso,so(t:) + it gr(f) dr + it J hr(f, y)N(ds x dy). So So

where Yso,so(f) is :Fso-measurable, and 9r(f) hr(f,y) are predictable processes for each fixed (f, y) E (-1, 1) x R. Suppose that, for any p ~ 2,

(7.7) sup E[lYso,so (f) IP] + sup E[ sup IYso,r(f)12] < 00. t"E( -1,1) fE( -1,1) so<r~l

Moreover, we suppose that there exists a predictable process 'T7t ( f) (so < t :S 1) such that, for any t E (so, 1]

(7.8) 19t(f)1 :S C( sup IYr_(f)1 + l'T7t(f) I),

so<r9

J Iht(f, y)IPv(dy) :S Cp(s:~~t IYr_(f)1 + l'T7t(f)I)P,

43

then we have

Remark. The proof of the inequality (7.9) is essentially due to !4}, although the argument in !4} is for a concrete semimarlingale.

Proof. We set TR := inf{t > s; IYs,tl ~ R}

with convention that inf0 = 1. By the condition (7.7), we have limR~oo P(TR < 1) = o. Put ~~ = Ys,tATw By the monotone convergence theorem, we get

Hence, if (7.9) holds for yR with a constant Cp which does not depend on R, then we complete the proof. We have (7.10)

I tATR IP I tATR! IP

I~~(€)IP::; C~(IYso,so(€)IP+ lso gr(€)dr + lso hr(€,y)N(dr x dy) ).

By Schwarz's inequality and (7.8), we have

t - R tAT -Put Mso,t = Iso I hr(€' y)N(dr x dy), and Mso,t = Iso R I hr(€, y)N(dr x dy). By the Ito formula, we have

IM!,tIP = a martingale with mean zero

+ 1:"" ! (1M"" + h",,(y)IP -IM",,-IP - ph""(y)IM",,IP-l) v(dy)ds

The mean value theorem yields that

I I Mso,r + hs,rg(y)IP -IMso,r-IP - phs,rg(y)IMso,rIP-ll

::; ~p(p - 1)lhso ,rg(y)/2/Mso ,r + Ohso ,rg(y)/P-2

::; C:(/hso,r/2/Mso,r/P-2 + /hso,r/P),

44

Using Doob's inequality, we have

In the case p = 2, one can deduce the assertion from (7.10), (7.11), (7.12), (7.8), and Gronwall's lemma. In the case p > 2, it follows from (7.8) that, for r ::; t 1\ T R

and that

Hence, Holder's inequality yields

Therefore, the right hand side of (7.12) is bounded by

Cp E[lYso,so(E)/P + it (/17r(E)/P + sup /~:,rl/P) dr] So so<rl$r

Hence, (7.10), (7.11), and Gronwall's lemma also show (7.9) for p > 2. 0

45

Lemma 7.0.2. For any p ~ 2, there exists a constant Cp > 0 such that the following inequalities hold

(7.13)

(7.14)

E[ sup IXs(t) - Xs(tl)IP] ~ Cplt - tllP O:::;s:$1

Remark 5. From (7.14), one can check that

By applying Lemma 1.1 in Fujiwara-Kunita !4J, we know that there is a version X;(t) of Xt(t) such that t -+ X;(t) is continuously differentiable.

yt(t,td = lot (b(Xs_(t)) - b(Xs-(t))) ds

+ it (w(Xs_(t), y) - tla(Xs_(t), y)) N(ds x dy) 0+

Note that

Ib( Xs-(t)) - b{ Xs-(tl)1 ~ sup Ib(l)(x)11 Ys-{t, tdl, x

and

! Iw(Xs-(t), y) - tla(XS-(t), y)IPv(dy)

~ Cp [It - tllP ! la(Xs-(t), y)IPv(dy) + ! la(Xs-{t), y) - a{Xs-(tl)' Y)IPV(dY)]

~ Cp ( It - tIl (1 + IXs-I)P + IYs-(t,tl)IP ),

where we used (7.5). By applying Lemma 7.0.1 with So = 0 and 'f/so,s(y) = It - tIl (1 + IXs-(t)I), we have (7.13).

46

Next, by using Lemma 7.0.1, we shall also show (7.14). Note that

It{E,El) It{E,E2)

E - El E - E2

= it (b{ Xs_{E)) - b{ Xs_(E) _ b( Xs-(E)) - b( Xs_{E) )) ds o E - El E - E2

+ 1: J B(E, Eb S, y) N(ds x dy),

where B( E, El, S, y) is given by

W(XS_(E), y) - Ela(Xs_(E), y) W(Xs_(E), y) - Ela{Xs_{E), y) E - El E - El

We define 'fJs = (1 + I Y..;~:~E2) I) IYs- (El' E2) I· Then, the inequality (7.13)

shows that E[sUPO:::;s91'fJsIP] ~ lEI - E2lp .One can write

Because b has bounded derivatives, we have

In a similar way, we have

47

Hence, we have

J 13(E, E1, s, y)IP v(dy) :::; Cp J 1 a(Xs_(E1), y) - a(Xs_(E2), y)IP v(dy)

+ Cp J 1 a(Xs_(E), y) - a(Xs_(E1), y) - a(Xs_(E), y) - a(Xs_(E2), y) I

P v(dy) E - E1 E - E2

:::; Cp (I Ys-(E, E1) _ Ys-(E, E2) IP + l'T]s-IP) ,

E - t1 E - E2

where, in the last inequality, we have also used

J 1 a(Xs-(Ed, y) - a(Xs_(E2), y)IP v(dy)

:::; (s~p J la(1)(x, Y)IPV(dY)) IYs-(Eb E2)IP :::; Cp'T]s.

By applying Lemma 7.0.1, we get the result. o

As we mentioned in Remark 5, we can take a modification of X(t) such that E --+ X(t) is continuously differentiable. We denote by the same symbol X(E) the continuously differential version. We denote also X~O)(E) = Xt(E) and X(1)(E) = d~lE). Then, one can check that X(1)(E) satisfies

XP)(E) = lot b(l)(Xs_(E))X!~ds + lot a(Xs_(E), y)N(ds x dy)

+ E lot J a(l)(Xs_(E), y) X!~(E)N(ds x dy).

Further, applying the same argument to the process X(l)(E), we see that E --+ X(l)(E) has continuously differential version. Repeating the same argument inductively, we have:

Theorem 7.0.1. There exists a version of X(E) such that E --+ Xt(E) zs infinitely differentiable. Moreover, for n = 2,3 ... , x(n)(E) = tE:X(E) is given by formal n-times derivative of X(E) with respect to E:

(7.16)

X~n)(E) = lot !: b(Xs_(E))ds + lot J !: ( W(Xs_(E), y) ) N(ds x dy).

48

Remark. The stochastic differential equation is defined by induction. More precisely, the stochastic differential equation (7.16) has the unique solution if h X (O) X(n-1) . t e processes t , ... , t are gwen.

Lemma 7.0.3. For any n, any €, and any t, xin)(€) is in Doo. Further, we have

(7.17) sup sup /xin )(€)/ < 00. 0<t9 t:E(-l,l) k,p

Proof. To prove this, we shall use induction with respect to n. In the case n = 0, the claim is true because of (7.6). Let n ~ 1 be fixed. Suppose that for each 1 ~ n, and for each non-negative integer k and for p ~ 2, (7.17) holds. Under this assumption, we shall prove

(7.18) sup /xin+!)(€)/ < 00. t:E( -1,1) k,p

We set

~+1 Fsl(€) = dEn+! b(Xs(€)) - b(l)(Xs_(€))x(n+!)(€)

~+1 F;(€, y) = dEn+! (m(Xs(€), y)) - m(l) (Xs-(€), y)x(n+l)(€).

The variable F'L ( €, y) can be written by a linear sum of random variables such as

(7.19) a(l)(Xs(€),y) (Xil))h ... (xin))'n (l = 0, ... ,n) or

m(l') (Xs(€) , y) (Xil))h ." (xin))'n (l' = 2, ... , n + 1),

where 0 ~ h, ... , In ~ n. Hence, by the assumption of the induction, we have

J IDk F2(€ ) IP

(7.20) sup sup esssupE[ U ( )' Y lI(dy)] < 00. t: 0<s9 uEA(l)k 'Y (J

In a similar way, one can check that

(7.21) sup sup IFl(S, €)Ik,p < 00. t: 0<s9

49

Pick u = (81, Yl) E A(l). Note that Duxin+1) = 0, if 81 > t. Hence, to prove

(7.18) for k = 1, we have to see that

where

sup sup esssup 1, 1 < 00.

II Ds y Xt(f) II

€ 0<t9 (SI,yI)EA(l)n(O,tjxR IYll p

Ys1,t = t (Du(FL(f) + b(1)(X2_(f))))X!~+1)(f) d8

lSI + 1: / (Du(F'L(f, y) + a(1)(X!~(f), y) ))X!~+1)(f) N(d8 X dy).

Recall that F'L(f, y) can be written by a linear sum of random variables such as (7.19). We use the convention that D~G = G. Then, using Lemma 7.7 with 80 = 8 and

17t = (IDuxiO) I + L ID!o XiO\f)'" D!n xin) (f) I) Ixin

+1) (f) I,

we have

kjE{O,I} l~kO+···kn~n

where, in the last inequality, we used the assumption of the induction. By Assumption 1, we have

50

It follows from (7.3), (7.4), and (7.20) that

(7.22) sup IIF;1 (€, Yl) + a(1)(X~~)(€), Yl)X~~+l)(€)llp ~ CIY11· 81

Hence, using Lemma 7.0.1 with So = Sl, 'fit = Ys1,t and YsQ ,8Q = F;1(€,Yl) + a(1)(X~~)(€), ydX~~+1)(€), we have

sup E[ sup 1·,t1,t(Yl, €) IP] e 81 <t:9

~ Cp II F;1 (E, yd + a(l)(X!~)(€), Yl)X!~+1)(€)II~1 + E[!t IYs1,1IP dr] 81

~ CpIYlIP.

This implies that sUPeSUPt Ixin+1)(€)ll,P < 00. Repeating this argument, one

can get the result. 0

Theorem 7.0.2. The Taylor expansion

is the asymptotic expansion.

Proof. Taylor's formula yields that

Therefore, it follows from Lemma 7.0.3 that

< C sup IXm+l(E')1 €m+l. - m 1 k,p

k,p E'

o

We define F(€) := X1(E)~X1(0), then F(€) has the asymptotic expansion (n+1)( )

:E~=o En fn with In = Xtn+l)!o . We shall give a sufficient condition that F( €)

51

satisfies the uniformly non-degenerate condition. Let Z: (s < t) be a solution to the following linear stochastic differential equation;

Zt(f) = 1+ it b(l) (Xr-(f)) Zr-(f) dr+f it! a(l)(Xr_(f), y) Zr-(f) N(dsxdy).

This equation is given by the derivative of the stochastic differential equation

Xs,t(f) = x + it b(Xs,r-(f)) dr + fit! a(Xs,r-(f), Y)(f) N(ds x dy).

with respect to the initial value x. Put '¢s(f) = Zs,l(f)a(Xs(f)). Then, as in Picard [12], one can check that for any p ~ 2 there is a constant C > 0 such that for any u = (s, x) E A(1)

(7.23) sup IID(s,x)F(f) - Gt(f)xllp :::; Oy(uto E

where ro is given in condition (7.4).

Theorem 7.0.3. Suppose that limsuPE~osuPtE[IXt(f)I-P] < 00 for any p ~ 2, and that there is a positive constant c > 0 such that

la(x)1 ~ clxl, lim infinf(1 + W(l)(X, y)) ~ c E~O x,y

then F(f) := Xl(E)~Xl(O) satisfies the uniformly non-degenerate condition.

Proof. We use Theorem 5.2.2 with '¢t(f) = Zs,l(f)a(Xs(f)) defined above. Because of (7.23), all we have to do is to show that the condition (5.2) holds. Thanks to Jensen's inequality, we have

II( t ('¢t(f))2dt)-lilp :::; II( fll'¢t(f))21-1dtllp :::; sup II,¢t1(f)II2p. 10 10 t

We define t! (W((l)(Xr_(f), y))2 Ws,t(f) = -Ks,t(f) + 1s 1 + W(1)(X

r_(f),y)N(dr x dy),

where Ks,t(f) = fst b(l) (Xr_)dr + f fst f a(1)(Xr_(f), y)N(dr x dy). Then, by the assumption, Ws,t(f) is well-defined for each f > 0 small enough. It follows from Theorem 63 in [14] that Z;,l(f) (s < t) satisfies

Z:;](f) = 1 + it Z~i_(f)dWs,r(f).

52

Hence, one can check that limsuPf-tosups Ilz;,t(E) lip < 00. From the condi­tion limsuPf-tosUPtE[lXt(E)I-P] < 00 and la(Xs(E)I ~ CIXs(E)I, the assertion follows. D

If a is uniformly non-degenerate, then we need not to assume the condition lim SUPf-tO SUPt E[IXt( E) I-P] < 00

Example 7.1. Suppose that the Levy measure v(du) is given by

v(dx) = l{x;lxl~R}C (l(_oo,o)(x)eGx + 1(0,00) (x)e- MX) Ixl-(HY)dx,

where C, G, M are positive constants and Y < 2. This is known as CGMY­model in mathematical finance if R = 00. We suppose that 0 < Y < 2, then Condition 1 holds. Put gf(Y) = log(l + E(eY - 1)). Let 1 < R < 00 be fixed. Then, one can check that, for any y E [-R, R],

(7.24)

We define b(E) = J::R( eg·(y) - 1 - gf(y))v(dy), and consider the following process;

It(E) = (b - b(E))t + It J gf(y)N(dr x dy).

Then, by the Ita formula, we have

(7.25) Xt(E) := eLt(f) = 1 + b t Xr_dr + E tjR Xr_(eY -l)N(dr x du). 10 10 -R

On the other hand, for E > 0, the process It can be considered as a Levy process with characteristic exponent ~(e)

- 11og(Hf(eR

-1» ~(e) = logE[eieL1(f)] = b - b(E) + (eiey -1- iey)vf(dy),

log(Hf(e-R-1»

where vf(dy) = f+::-1 (v~) I ) dy. By Lemma 25.6 in Sato x=log( He-I (eY -1»

[16J, we see that E[lXt IP] = E[ePLt(f)] exists for any pER, and by Theorem 25.17 in Sato [16J, we get

limsup sup E[IXtIP] = lim sup sup E[ePLt(f)] f-tO 0<t9 f-tO 0<t9

:S li,!,-!,~p exp {Ip(b - b( f»~ + I (eI'" - 1 - py)v, (dy)I } < 00

53

where, in the last inequality, we used (7.24) after changing variable y = log(1+c1 (eX -l)). Hence, the stochastic differential equation (7.25) satisfies the condition in Theorem 7.0.3. We cannot choose R = 00; Xt(E) does not belong to LP for large p if R = 00 . However, it seems important to study the asymptotic expansion of (7.25), if one want to know the property of geometric CG MY process.

54

Chapter 8

Appendix

8.1 Closability of D

The purpose of this section is to prove the closability of D. Recall that the operator D is closed on each chaos en. Now, let Fn := E~o Ik(J~n») be a sequence which is contained in the domain of D. Suppose that there is a random variable G such that Fn -+ G, and suppose also that E[fA(l) IDuFn­Zul 2 )'(du)] -+ O. Then, we have

E[lIk(J~n») - h(f~m»)12] kl r If~n)(a) - f~m)(a)12)'(da) JEk

< Ell 1 If/n) (a) - f/m)(a)12)'(da) l=O El

E[lFn - Fm12] -+ 0,

and have also

E[ r IDulk(f~n») - Duh(J~m»)12)'(du)] J A(l)

= k2E[ r IIk_l(J~n)(.,u)) - Ik_l(f~m)(.,u))12)'(du)] J A(l)

~ E[ r IDuFn - DuFml2 )'(du)] -+ o. J A(l)

56

Because the restriction of D on Wiener chaos is closed, there exists a function 11k E (L2(A(1)))0n such that

lim E[lIkUkn)) - Ik(11k) 12] + E[ r IDulkUkn)) - Dul k(11k)1 2A(du)] = 0 n-+oo J A(l)

For any m, we have

and

t kk! r A(du) r I 11k (a, u)12 A(da) k=O J A(l) J Ek-1

~ lim sup f kk! r A (du) r Ifkn) (a, U)12 A(da) < 00. n-+oo k=O J A(l) J Ek-1

This implies that G := E~o Ik(11k) is in L2(n) and E[fA(l) IDuGI2 A (du)] < 00. We may suppose that G and Zu can be written by G = E~o Ik(gk) and Zu = E~o Ik(hk(·, u)), respectively. Then, one can easily check tha~ Ik(gk) = Ik(11) and that h(hk(-,u)) = Dul k+1(11k+1). This means G = G and Zu = DuG in L2(n x A(l)). Therefore, the operator D : L2(n) 1-7

L2(n x A(l)) is closed.

8.2 Proof of Theorem 3.2.3

We divide the proof of Theorem 3.2.3 into several steps. Stepl. For simplicity, we denote Z~p = z~j)(e) and D~(n)(G) = D~~~~(G). Put Zy),l := ZY)lA(l)c(U). We also define D~~n)(G) as in Lemma 3.1.2. By Lemma 3.1.2, all we have to do is to show that for any sufficiently large l

IID~~n)(G)1I2 ~ C(l + I e 12)-(1-2P)~ IGln,q,

where C does not depend on l. The assumption of Theorem 3.2.3 yields that for any k, j and for any p ;::: 2

(8.1) IZu),11 _1 < IZU)I _1. < Ck .( II: Ir(11: I-~) )-1 k,p,1 ~ I 7! - k,p,1 ~ I f3 - ,PJ "" ...

where Ck,p,j is a constant which depends on p, k, j.

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Lemma 8.2.1. Under the notation above, we have

where Uj = (U1," .,Uj) and Sn = {(UI, ... ,un) E En; Ui =/:. Uj ifi =/:'j}.

Remark. See proof of Lemma 2.5 in Picard {13}.

From Lemma 8.2.1, we can write

where,

Zler) = (fI Z~),l OC;'i) (fI ZM),1 OC~) 3=1 3=1

with r = (U1, ... , Un, VI, ... , vn). We divide the region Sn x Sn as follows; for J1 = {jt, ... ,j~},J2 = {jf, ... ,j!} C {1,2, ... ,n}, we define

Uil = Vi2 if and only if i1 = j: } and i2 = j~ for some 1 ::; q ::; m .

Then we can write

where IJI is the cardinal number of the set J. Let J1 = {jt, ... ,j~}, J2 =

{j?, ... ,j~} be fixed. Let us estimate the expectation of III ZI(r)N(dr). Jl,J2

Step 2. Put m = IJ11 and k = n - m. Define {it, ... , in = {I, ... , n} n Jf and {i?, ... , in = {I, ... , n} n J2. We denote 0"1 = (Uj}, ... , uH,.) and 0"2 =

(Ui1, ... , Ui1 , Vi2, ••• , Vi2 ). Then, on the set tl.Jl h, we can regard Zl (r) as the 1 k 1 k '

process parametrized by (0"1,0"2) E Sm+2k. We set

58

Then, we can write

(8.2)

Note that N(du) x N(du) = N(du). Recall that the measure Ao(du) is given by (3.2). By applying Lemma 2.4 in Picard [12], we have

where c = JA(l) ,2(U)A(du) , a~ is extracted from aI, and the sum is obtained by the formulas (2.10) and (2.8). The formula (2.10) and the fact that

-1

,(u) ~ 1 on A(I e IT) yields

where VI, v2, and V are extracted from (aI, a2) and the sum is given by the formula (2.10). Step 3. We shall estimate

(8.3)

59

For any r > 2, we take r' > 1 such that ~ + ~ = 1. By applying Holder's r r

inequality, (8.3) is bounded by

(8.4)

r II DVI G 0 c~n DV2 G 0 c;n IIII DvZI(al' a2) II Ao(al)Ao(da2) J Sm+2k 'Y(VI) 'Y(V2) ~ 'Y(v) r'

::; r IIDvIGoc~nIIIIDv2Goc;nIIIIDvzl(al,a2)11 AO(al)AO(da2). J S.,.+2k 'Y(vd r 'Y(V2) r 'Y(v) r'

It follows from (2.1) that for any random process Xu

Xu 0 c; = Xu a.e.-Ao(du)A(dT).

Hence, it follows from (8.1) and (2.10) that,

liD Zl(a a) II ( 1 )-2n s~p esssup -1 v 'Y(V) , 2 ,::; C Ie 1r(1 e I-p) ,

ur,u2EAm+2k (I E 17) r

where C > 0 does not depend on l. Note that the region of the integral in (8.4) is A(I e Ifr)2k+m, because Z!P = 0 on A(I e Ifr). For j = 1,2, applying HOlder's inequality, we have

1 IIDvoGl1 sup 1 _J_ Ao(dVj)

IGln,r=1 A(IEI-iT) 'Y(Vj) r

[1 IDvoGlr ]~ ( 1 )(I-~)IVjl

::; sup E _1 _(J) Ao(dVj) AO A(I e I-p) IGln,r=1 A(I E I iT) l' Vj

::; r( Ie I-fr )(I-~)Ivjl.

where IVjl (j = 1,2) is the length of Vj (j = 1,2), respectively. Remark that (2.9) and definition of all a2 imply that VI and V2 have no same component, further VI and V2 have to be extracted from a2. Hence, IVII + IV21 ::; 2k holds. Recall that r > 2. The right hand side of (8.4) is bounded by

II J II Dv~~ II Ao(dvj)· ess sup II DvZI(al, a2) II, r(1 e l-fr)m+2k-(lv11+lv21) j=1,2 1'( J) r Ur,U2 'Y(v) r

::; CI e 1-2n r(1 e l-fr)-2n+m+2k-(lvll+lv21)+(I-~)(lvll+lv21) ::; CI e 1-2n r(1 e l-fr)_n+k_lvll~lv21 ::; CI e 1-2n r(1 e I-frtn ::; CI e 1-2(1-~)n,

60

where, in the last inequality, we have used r-2) in Condition 1 in Chapter 2. The constant C does not depend on l > O. This completes the proof.

61

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