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The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density Vlad BALLY , Denis TALAY

N˚ 2675 Octobre 1995

PROGRAMME 6

ISSN 0249-6399

apport de recherche

The Law of the Euler Scheme for Stochastic Dierential Equations : II. Convergence Rate of the Density Vlad BALLY  , Denis TALAY  Programme 6  Calcul scientique, modélisation et logiciel numérique Projet Omega Rapport de recherche n2675  Octobre 1995  30 pages

Abstract:

In the rst part of this work [4] we have studied the approximation problem of IE f (XT ) by IE f (XTn), where (Xt ) is the solution of a stochastic dierential equation, (Xtn) is dened by the Euler discretization scheme with step Tn , and f ( ) is a given function, only supposed measurable and bounded. We have proven that the discretization error can be expanded in terms of powers of n1 under a nondegeneracy condition of Hörmander type for the innitesimal generator of (Xt ). In this second part, we consider the density of the law of a small perturbation of XTn and we compare it to the density of the law of XT : we prove that the dierence between the densities can also be expanded in terms of n1 . The results of this paper had been announced in special issues of journals devoted to the Proceedings of Conferences: see Bally, Protter and Talay [2] and Bally and Talay [3]. AMS(MOS) classication: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05. AMS(MOS) classication: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.


Key-words: Stochastic Dierential Equations, approximation, Malliavin calculus. (Résumé : tsvp)

Université du Maine, et Laboratoire de Probabilités, Université Paris 6, 4 Place Jussieu, 75252 Paris Cédex 05, France. E-mail : [email protected] .  INRIA Sophia-Antipolis, 2004 route des Lucioles, B.P. 109, 06561 Valbonne Cedex, France. E-mail : [email protected] . 

La loi du schéma d'Euler pour les équations diérentielles stochastiques: II. Vitesse de convergence de la densité Résumé :

Dans la première partie de ce travail [4] nous avons étudié l'approximation de

IE f (XT ) par IE f (XTn), où (Xt ) est la solutio d'une équation diérentielle stochastique, (Xtn) est déni par le schéma d'Euler de pas Tn , et f ( ) est une fonction seulement supposée mesurable


bornée. Nous avons prouvé que l'erreur de discrétisation peut être développée en puissances de 1 n sous une hypothèse de type Hörmander pour le générateur innitésimal de (Xt). Dans cette seconde partie, nous considérons la densité de la loi d'une petite perturbation de n XT et nous la comparons à la densité de la loi de XT : nous prouvons que la diérence entre les densités peut elle aussi être développée en puissances de n1 . Les résultats de cet article ont été annoncés dans des numéros de journaux consacrés aux Actes de Conférences: voir Bally, Protter et Talay [2] et Bally et Talay [3]. AMS(MOS) classication: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05. Classication AMS(MOS): 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.

Mots-clé : Equations diérentielles stochastiques, approximation, calcul de Malliavin.

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 3

1 Introduction Let (Xt ) be the process taking values in IRd solution to

Xt = X0 +

t

Z

0

b(Xs )ds +

Z

0

t

(Xs )dWs ;

(1)

where (Wt ) is a r-dimensional Brownian motion. The problem of computing the expectation IE f (Xt ) on a time interval [0; T ] by a Monte Carlo algorithm appears in various applied problems; some of them are listed in [4]. The algorithm consists in approximating the unknown process (Xt ) by an approximate process (Xtn ), where the parameter n governs the time discretization; that process can be simulated on a computer, and a simulation of a large number M of independent trajectories of Xtn provides the following approximate value of IE f (Xt ): 1

M

n

M i=1 f (Xt (!i )) : X

The resulting error of the algorithm depends on the choice of the approximate process and the two parameters M and n. In [11] the choice of the Euler scheme instead of a more sophisticated scheme to approximate (Xt ) is justied. Thus, we consider: X0n = X0 ; (2) n n + b(X n ) T + (X n )(W(p+1)T=n ? WpT=n ) : X(p+1)T=n = XpT=n pT=n pT=n n (

For

pT n

n  t < (p+1)T n , Xt is dened by

n + b(X n ) t ? pT + (X n )(W ? W Xtn = XpT=n t pT=n ) : pT=n pT=n n 



When X0 = x (resp. X0n = x) a.s., we write Xt (x) (resp. Xtn(x)). The eects of n on the global error of the algorithm can be measured by the quantity jIE f (XT ) ? IE f (XTn)j : (3) This error can be expanded in terms of powers of n1 : see Talay and Tubaro [11] for smooth f 's without any assumption on the innitesimal generator of (Xt ) and for the numerical interest of the result (i.e. the justication of Romberg extrapolations which exponentially accelerate the convergence rate with a linear increase of the numerical cost). Similar results hold when (Xt ) is the solution of a Lévy driven stochastic dierential equation, see Protter and Talay [10]. In Bally and Talay [4] the same expansion has been established for only measurable and bounded functions f 's under a uniform nondegeneracy condition of Hörmander type on that generator (see below for a more precise formulation). RR n2675

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Vlad BALLY , Denis TALAY

In this paper our objective is the following. Let UL  IRd be the set of points for which Hörmander's condition involves Lie bracketts of length less or equal to L. For x 2 UL the law of XT (x) has a density pT (x;
) with respect to Lebesgue's measure. We want to approximate this density by the density p~nT (x;
) of the law of of a small perturbation of XTn (x). More precisely, for x and y in UL , one has pT (x; y) ? p~nT (x; y) = ? 1 T (x; y) + 12 RTn (x; y)

n n for some function T (x; y) independent of n and some remainder term RTn (x; y) which

satisfy an exponential inequality of the type

jT (x; y)j + jRTn (x; y)j 

2 k x ? y k K (T ) : T q VL(x)q0 VL (y)q00 exp ?c T !

(the functions VL(
) and K (
) are dened below). The above expansion and the exponential bounds for T (x; y) and RTn (x; y) give a local information on the approximation of the law of XT (x) by the law of X~Tn (x): without any global nondegeneracy assumption but under the hypothesis that x 2 UL and A  IRd is a Borel set whose boundary is a subset of UL (neither A nor Ac is supposed included in UL ) we prove that IP [X (x) 2 A] ? IP [X~ n (x) 2 A] = ? 1  (x; A) + 1 Rn (x; A) T

T

n

T

n2

T

where the functions jT (
; A)j and jRTn (
; A)j can be bounded exponentially from above. During the very last days of the redaction of the present paper, the authors have received a paper by Kohatsu-Higa [7] which also deals with the approximation of pT (x; y) for x 2 UL . This density is approximated by

G ?y n where G is a standrad Gaussian vector independent of W1 and  (
) denotes a Gaussian density of mean 0 and of covariance matrix  2 Id(IRd). Kohatsu-Higa shows that, for any   1, there exists a constant C () such that C () supd jIE n? (XTn (x) ? y) ? pT (x; y)j  n : y2IR 

IE n? XTn(x) +



It is clear that this estimate and our results are of dierent nature. The organization of the paper is the following: in Section 2, we state and comment our main results; in Section 3, we prove these results, admitting technical estimates proven in Section 4; these estimates require a modication of a result due to Kusuoka and Stroock concerning the derivatives of pT (x; y): this work is done in Sections 5 and 6. INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 5

Notation. In all the paper, given a smooth function (
) and a multiindex  of the form

 = (1 ; : : : ; k ) ; i 2 f1; : : : ; dg the notation @z (t; z;  ) means that the multiindex  concerns the derivation with respect to the coordinates of z , the variables t and  being xed. When we write @ (t; z ) it must be understood that we dierentiate w.r.t. the space variable z only. When = ( ij ) is a matrix, ^ denotes the determinant of , and j denotes the j ? th column of . When V is a vector, @V denotes the matrix (@i V j )ij . We will use the same notation K (
), q, c, , etc, for dierent functions and positive real numbers having the common property to be independent of T and of the approximation parameter n: typically, they will only depend on L1 -norms of a nite number of partial derivatives of the coordinates of b(
) and (
) and on an integer L to be dened below. As usual, we denote by IPz the law for which X0n = X0 = z a.s. and we denote the corresponding expectation by IEz . In all the paper, we reserve the letters x and y for elements of a set UL dened below.

2 Main results 2.1 Density and local density. Consider the stochastic dierential equation (1). In all the paper we suppose:

(H) The functions b(
) and (
) are C 1(IRd) functions whose derivatives of any order are bounded.

Denote by A0; A1 ; : : : ; Ar the vector elds dened by d

A0(
) =

X

Aj (
) =

X

i=1 d

i=1

bi(
)@i ; ij (
)@i ; j = 1; : : : ; r :

For multiindices  = (1 ; : : : ; k ) 2 f0; 1; : : : rgk , dene the vector elds Ai (1  i  r) by induction: A;i = Ai and, for 0  j  r, Ai(;j) := [Aj ; Ai ]. RR n2675

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Vlad BALLY , Denis TALAY

For L  1, dene the quadratic forms

VL (; ) := and set Denote by UL the set

r

X

X

j=1 jjL?1

< Aj ( );  >2

VL ( ) = 1 ^ kinf V (; ) : k=1 L

(4)

UL := f ; VL ( ) > 0g :

(5)

Kusuoka and Stroock (Corollary 3.25 in [8]) have shown: for any integer L  1 and any x 2 UL , the law of XT (x) has a smooth density pT (x;
); besides, for any integers m; k, for any multiindices  and such that 2m + jj + j j  k, there exist an integer M (k; L), a non decreasing function K (
) and real numbers c; q depending on L; T; m; k; ; and on the bounds associated to the coecients of the stochastic dierential equation and their derivatives up to the order M (k; L), such that the following inequality holds1:

@ m @ x @ z p (x; z )  K (T ) exp ?c kx ? z k2 ; 80 < t  T ; 8z 2 IRd ; 8x 2 U : L @tm  t tq VL(x)q+2q=L t

!

(6)

A complementary result also holds whose proof is postponed to Section 5.

Proposition 2.1 Assume (H). Let L be such that UL is non void. Then there exists a smooth function

(t; z; y) 2 (0; T ]  IRd  UL ! qt(z; y) such that, for any measurable and bounded function (
) with a compact support included in the set UL , one has IEz (Xt ) = ( )qt (z;  )d : (7) Z

Supp()

Let L  1, m be arbitrary integers and let ; be multiindices. There exist positive constants , c and there exists an increasing function K (
) such that, for any 0 < t  T ,

8y 2 UL ;

8z 2 IRd

@ m @ z @ y q (z; y)  K (T ) exp ?c k z ? y k2 : ; @t m  t (tVL (y)) t



!

(8)

2.2 The perturbed scheme. If the uniform ellipticity condition

(H1) 9 > 0 ; k(t; x) (t; x)k   ; 8(t; x) 1

In the statement of Kusuoka and Stroock, the constants j , n (L) are equal to 0 under (H).

INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 7

holds, then the law of XTn (x) has a density pnT (x;
) w.r.t Lebesgue's measure. It may be false even if there exist L > 0 and  > 0 such that VL ( )   > 0 ; 8 . That leads us to consider a small perturbation of XTn whose law has a density. In the whole paper, we refer to the following

Denition 2.2 Let 0(
) be a smooth and symmetric probability density function with a compact support in (?1; 1). For  > 0 and  2 IRd we dene:  ( ) :=

d

0( i =) :  i=1 Y

(9)

We now dene a new approximate value of XT (x), denoted by X~Tn (x): let Z n be a IRdvalued random vector independent of (Wt ; 0  t  T ) whose components are i.i.d. and whose law is 1=n( )d ; we set:

80  t < T ; X~tn(x) := Xtn(x) ; X~Tn(x) = XTn(x) + Z n : We denote by p~nT (x;
) the density of the law of X~Tn (x) w.r.t. Lebesgue's measure.

(10)

2.3 Convergence rate for the density. In [4] we have proven that, if inf z2IRd VL (z ) > 0 for some integer L and if the functions b(
) and (
) satisfy (H), then for any measurable and bounded function f (
),

IEx f (XT ) ? IEx f (XTn ) = ? Cf (nT; x) + Qn(f;n2T; x) ;

(11)

the terms Cf (T; x) and Qn(f; T; x) having the following property: there exists an integer m, a non decreasing function K (
) depending on the coordinates of (
) and b(
) and on their derivatives up to the order m, and a positive real number q such that (12) jCf (T; x)j + supnjQn(f; T; x)j  kf k1 KT(Tq ) (in fact, the estimate given in [4] is slightly dierent: the simplied version (12) takes the boundedness of b(
) and (
) into account). To get an expansion for pT (x; y) ? p~nT (x; y), it is natural to x y, to choose f ( ) =  (y ?  ) and to make  tend to 0. But the above result is not sucient since, when  tends to 0, (k f k1 ) tends to innity. Nevertheless, if F (
) is the distribution function of the measure f ( )d , the sequence (k F k1 ) is constant: this gives the idea of proving inequalities of type (12) with k F k1 instead of k f k1 when f (
) has a compact support, F (
) being the distribution function of the measure f ( )d . Before stating our main results we need to introduce some denitions.

RR n2675

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Vlad BALLY , Denis TALAY

2.4 Denitions. Denition 2.3 Let (aij (t; x)) denote the matrix (t; x) (t; x). Let (
;
) be a function of class C 4([0; T )  IRd). We dene the dierential operator U

by

d

d

U (t; z) := 12 bi(z)bj (z)@ij (t; z) + 12 bi (z )ajk (z )@ijk (t; z ) i;j=1 i;j;k=1 X

X

1 d ij kl 1 @2 a (z )a (z )@ijkl (t; z ) + 2 (t; z ) 8 i;j;k;l=1 2 @t d 1 d ij @ @ (t; z ) : @ a (
) @t ij + bi (
) @i (t; z ) + @t 2 i;j=1 i=1 X

+

X

X

(13)

Denition 2.4 We set u(t;  ) := IE f (XT ?t ) = PT ?tf ( ) ; 0  t  T :

(14)

Proposition 2.1 shows that, if f (
) is a measurable and bounded function with a compact support included in a non void set UL dened as in (5), the function u(
;
) is of class C 1([0; T )  IRd). We set (t;
) := U u(t;
) : (15)

Denition 2.5 Let L 2 IN ? f0g such that the set UL is non void. For any measurable bounded function f (
) with a compact support included in UL we set f;L = 2Supp(f) inf VL ( ) > 0

(16)

and we denote by df ( ) the distance of  to the support of f (
). We also denote by Ef;L( ) any function dened on UL of the form 2 Ef;L() := T QKq (VT )()q0 exp ?c df T() ; f;L L !

(17)

for some strictly positive constants c, q, q0 , Q and some positive increasing function K (
).

2.5 Statements. Theorem 2.6 Assume (H). Let L 2 IN ?f0g be such that UL is non void and let x 2 UL. Let f (
) be a measurable and bounded function with a compact support included in UL . Let F (
) denote the distribution function of the measure f ( )d . INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 9 Let (
;
) be dened as in (15) and for x in UL set

T Tf (x) := 0 IRd pt(x; z ) (t; z )dzdt Z

Z

:

(18)

The perturbed Euler scheme (10) satises: there exists some function Ef;L (
) (for some strictly positive constants Q, c, q, q0 and some increasing function K (
)) and for each n > df2(x) , there exists RTn;f (x) such that (19) IEx f (XT ) ? IEx f (X~Tn ) = ? 1 Tf (x) + 12 RTn;f (x)

n

and

n

(20) jTf (x)j + jRTn;f (x)j k F k1 Ef;L(x) : The function K (
) entering in the denition of Ef;L (
) depends on the Lm(IRd ) norms (for some integer m) of a nite number of partial derivatives of the function 0(
).

Under (H1), (19)-(20) also hold for X n instead of X~ n and any bounded measurable f (
) with a compact support.

Corollary 2.7 Assume (H). Let L 2 IN ? f0g be such that UL is non void and let x and y be in UL , so that Set

VL (x) ^ VL (y) > 0 : Z

T

(21)

Z

pt(x; z )(U qT ?t(
; y))(z )dzdt (22) where the operator U is dened as in (13) and the function qT ?t(
; y) is dened as in (7). There exists a non decreasing function K (
), there exists some strictly positive constants c, q, q0, q00 and for each n > kx?2 yk , there exists a function RTn (x; y) such that the perturbed T (x; y) :=

0

IRd

Euler scheme satises

with

pT (x; y) ? p~nT (x; y) = ? n1 T (x; y) + n12 RTn (x; y)

(23)

K (T ) k x ? y k2 : exp ? c (24) T q VL (x)q0 VL(y)q00 T The function K (
) depends on the Lm (IRd) norms (for some integer m) of a nite number of partial derivatives of the function 0(
). Under (H1), (23)-(24) also hold for all (x; y) and for pnT (x; y) instead of p~nT (x; y).

jT (x; y)j + jRTn (x; y)j 

!

Theorem 2.6 and Corollary 2.7 cannot be seen as extensions of (11) which holds for the Euler scheme itself and for unbounded coecients b(
), (
). Nevertheless, the expansion (23) can be used to get a result similar to (11) when VL (
) is bounded below by a strictly positive constant uniform. Even weaker assumptions are admissible as shown by the following proposition. RR n2675

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Vlad BALLY , Denis TALAY

Proposition 2.8 Assume (H). L et A be a Borel set such that @A (the boundary of A) is included into a non void set UL for some integer L  1 (neither A nor Ac is supposed included in UL ). Let x 2 UL . Set Z TZ p (x; z )U (PT ?t llA )(z )dz : T (x; A) := d t 0

Then

n

A) + RT (x; A) : Px [XT 2 A] ? Px X~Tn 2 A = ? T (x; n n2

(25)

jT (x; A)j + jRTn (x; A)j  KT(QT ) E llA;L(x) :

(26)

h

Besides,

IR

i

Proof. Since UL is an open set and @A is included in UL one can nd a smooth function  (
) such that  (
) = llA (
) on ULc . As llA =  + ( llA ?  ), the result follows from Theorem 2.6 applied to f (
) = llA ?  (
) (since the support of f (
) is included in UL by construction) and from Theorem 1 of Talay and Tubaro [11] applied to the smooth function  (
) (the proof in [11] must be combined with the classical inequality (38) below to get the exponential in (26)).

3 Proofs of Theorem 2.6 and Corollary 2.7 For the sake of simplicity, in this section we use several technical estimates whose proofs are deferred in the next sections. Besides, we do not treat the restrictive case where (H1) holds, for which the arguments below can be used with some simplications.

3.1 Proof of Theorem 2.6. We recall the lemma 4.4 of [4]. The function (
;
) being dened as in (15) there holds

where

?2 2 nX IEx f (XTn ) ? IEx f (XT ) = Tn2 IEx k=0

kT ; X n + n?1 rn(x) ; n kT=n k=0 k !

X

(27)

rnn?1(x) := IEx f (XTn ) ? IEx (PT=nf )(XTn?T=n ) ; and for k < n ? 1, rkn (x) can be explicited under a sum of terms, each of them being of

the form

Z s Z (k+1)T=n Z s 1 2 \ n ] (X n )@ u(s ; X n )ds ds ds IEx '(XkT=n ) kT=n '  3 s3 3 2 1 kT=n kT=n  s3 # Z s Z (k+1)T=n Z s 2 1 [ y ' (Xs )@ u(s3; Xs3 )ds3ds2ds1 +'(XkT=n ) kT=n kT=n  3 kT=n h

(28) INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 11

where jj  6 and the '\ 's, '] 's, 'y's, '[ 's are products of functions which are partial derivatives up to the order 3 of the aij 's and bi's. Thus,

IEx f (X~Tn ) ? IEx f (XT ) = Tn

T

Z

IEx (s; Xs)ds T T 2 n?2 + 2 IEx kT ; XkT=n ? T IEx (s; Xs)ds n k=0 n n 0 2 n?2 n ? kT ; XkT=n + T 2 IEx kT ; XkT=n n k=0 n n 0

!

X

!

X

+ =:

nX ?2

k=0 TZT

n

0

Z

!!

rkn(x) + IEx f (X~Tn ) ? IEx (PT=n f )(XTn?T=n ) IEx (s; Xs)ds + An + B n +

nX ?2 k=0

rkn (x) + C n : (29)

We observe that the estimate (34) below ensures that 0T IEx j (s; Xs) jds is nite. First consider An: using Itô's formula and the estimate (34), we get R



T kT ; X I E x n kn?1 n kT=n ? !

X

T

Z

0



IEx (s; Xs) ds  kF k1 Ef;Ln2(x) :

(30)

Now we treat B n . For 1  k  n ? 2, one applies the expansion (29), substituting the function fn;k (
) := kT ;
!

n

to f (
). Set un;k (t; x) := PkT=n?tfn;k (
) and denote by n;k (t;
) the function dened in (15) with un;k (t;
) instead of u(t;
) and kT=n instead of T . Thus, for some functions g (
) 2 Cb1(IRd ) one has that, for t  kTn , n;k (t;
) =

X



g (
)@ PkT=n?t kT n ;
"

!#

:

There holds:

T 2 k?2 IE jT ; X n + k?1 rn;k (x) ; kT ; X n ? I E ; X = IEx kT x kT=n n kT=n n n2 j=0 x n;k n jT=n j=0 j !

!

X





where the rjn;k (x)'s are sums of terms of type (28) with un;k instead of u.

RR n2675

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Vlad BALLY , Denis TALAY

We use the inequalities (32) and (33) of the next section to upper bound the right side; we get: 2 n?2 n jBn j  Tn2 IEx kTn ; XkT=n ? IEx kTn ; XkT=n  kF k1 Ef;Ln2(x) : k=0 X

!

!

We proceed similarly to upper bound bound C n. That ends the proof.

n?2

X

k=0

(31)



rkn (x) and we apply (36) below to upper

3.2 Proof of Corollary 2.7. Fix y 2 UL and choose  small enough to ensure that the support of the function

 !  ( ? y) =

d

Y

1   i ? yi

i=1 

0

!



is included in the set UL (as VL(
) is continuous the set UL is an open set). Apply the estimate (19) with f (
) =  (
? y). Then F (
) is the cumulative distribution n (
) the functions function of the measure  ( ? y)d ; we denote by T;;y (
) and RT;;y appearing in the right side. There holds: n (x) : IEx  (X~Tn ? y) = IEx  (XT ? y) + 1 T;;y (x) ? 12 RT;;y

n

n

From Proposition 2.1 it is easy to check that T;;y (x) tends to T (x; y) when  tends to 0. Besides, as k F k1  1 it comes lim sup

!0



n (x)j jT;;y (x)j + jRT;;y



2  T q V (Kx)(qT0 V) (y)q00 exp ?c k x ?T y k : L L !

That ends the proof.

4 Upper bounds uniform w.r.t. n In this section we prove some technical propositions which have permitted us to upper bound the remaining terms of the expansion (29) (cf. the inequalities (31) and (30)).

INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 13

4.1 Statements. The following proposition was used to treat the term B n of (29).

Proposition 4.1 Let x, L 2 IN ? f0g, UL and the function f (
) be as in Theorem 2.6. Let g(
); g (
) be smooth functions in Cb1 (IRd; IR). Let  be a multiindex. Then there exist strictly positive constants c, q, q0, Q, a positive increasing function K (
) and the corresponding function Ef;L (
) such that jIEx [g(Xtn)@ u(t; Xtn)]j k F k1 Ef;L(x) ; 80  t  T ? Tn

(32)

and more generally, for any  in [t; T ? Tn ], for any function  (t;
) of the form

 (t;
) = g (
)@ [P?t (;
)] ;

one has

jIEx [ (t; Xtn )]j k F k1 Ef;L(x) ; 80  t    T ? Tn :

(33)

jIEx [g(Xt )@ u(t; Xt)]j k F k1 Ef;L(x) ; 80  t  T

(34)

jIEx [ (t; Xt )]j k F k1 Ef;L(x) ; 80  t    T :

(35)

Similar inequalities hold for the processes (Xt ) instead of (Xtn ); in that case, one may take 0  t    T : and

The next proposition was used to treat the term C n of (29).

Proposition 4.2 Assume the hypotheses of Proposition 4.1. Then jIEx f (X~Tn) ? IEx(PT=nf )(XTn?T=n)j k F k1 Ef;L(x) :

(36)

Before proving the two above propositions, we need to prove the two following technical lemmas, easy to obtain. The second one is interesting by itself.

4.2 Preliminary lemmas. Lemma 4.3 Let x 2 IRd and   IRd a closed set. The distance of x to  is denoted by

d(x; ). Let c > 0. For some strictly positive constants C0, C1 and C2 uniform w.r.t. n, T and  2 (0; T ], one has 2 n 2 (37) IEx exp ?c d(X; )  C0 exp C1 ? C2 d(x;) ; 80 <   T : !

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14

Vlad BALLY , Denis TALAY

Proof. If d(x; ) = 0, the right side of (37) is larger than 1 for C0 > 1, thus the inequality is true. If d(x; ) > 0, one splits the left side in two parts corresponding to the events A := [k Xn ? x k< 12 d(x; )] and Ac . On A one has that d(Xn; )  12 d(x; ), which gives n 2 IEx llA exp ?c d(X; ) "

!#

2  C0 exp ?C2 d(x;) : !

On the other hand, n 2 IEx llAc exp ?c d(X; )  IP k Xn (x) ? x k 21 d(x; ) : Since b(
) and (
) are bounded functions, we can use a standard inequality for certain "

!#





continuous Brownian semimartingales (obtained from a Girsanov transformation combined with Bernstein's exponential inequality for continuous martingales):

IP



k Xn(x) ? x k

1 d(x; )  C exp C  ? C d(x; )2 : 0 1 2  2 

!

(38)

Lemma 4.4 Under the hypotheses of Theorem 2.6, there exist some strictly positive constants Q, q, c and  such that

2 8T > t  0; 8z 2 IRd; j@z u(t; z)j  k F k1 (T ?K (t)TQ)  exp ?c dTf (?z)t : f;L !

(39)

Proof. If df (z ) = 0, the inequality is a consequence of (6) since Supp(f )  UL . We now x z 2 IRd such that df (z ) > 0. We rst observe that since VL (
) is a continuous function, one constructs a smooth function z;f;L : IRd ! [0; 1] such that

(a) if df (z)  pT ? t then z;f;L(y) = 1 for all y; (b) if df (z) > pT ? t then: (b1) z;f;L(y) = 0 if VL(y)  f;L y)  df2(z) , 2 or df ( (b2) z;f;L(y) = 1 if y 2 Supp(f ), (b3) IRd z;f;L(y)dy = 1 , (b4) for any multiindex with 1  j j  d, j@ yf;L(y)j  dfC(z) q  (T ?Ct) q= , where C R

2

is uniform with respect to df (z ) and f;L and where q is positive and depends on the dimension d only. INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 15

Since the support of f (
) is included in f ; z;f;L ( ) = 1g, one has (see Proposition 2.1 for the denition of the smooth function qT ?t(
;
)):

@z u(t; z )

= =

Z

@ z qT ?t (z; y)f (y )z;f;L (y )dy Supp(f)Z  (?1)d @y1 :::yd f@z qT ?t (z; y)z;f;L (y )gF (y )dy :

We now use the inequality (8) proven below. For some constants c and , for some increasing function K (
), it comes

8T > t  0 ; 8z 2 IRd; j@z u(t; z)j  k F k1 (TK?(Tt))

X

0j jd

Z

k z ? y k2 j@ z;f;L(y)j dy : exp ?c T ?t VL (y) !

From the denition of z;f;L (
), one has

j@ z;f;L(y)j > 0 )k z ? y k df2(z) and VL(y)  f;L 2 : Using the above conditions (a) or (b3) and (b4), we deduce (changing the denition of K (
) from line to line and remembering that f;L  1):

j@z u(t; z)j

2  k F k1   K(T(T?) t) exp ?c dTf (?z)t f;L 2 exp(c) exp ?c df (z ) lldf (z)pT ?t + 1 q T ?t df (z ) 2  k F k1 (T ?K t()TQ)  exp ?c dTf (?z)t : !

!

(

lldf (z)>pT ?t

)

!

f;L

4.3 Some recalls on Malliavin calculus. Now we are in the position to prove Proposition 4.1. The proof uses some material of [4] and some wellknown results which for convenience we rst recall in this subsection. We refer to Nualart [9] for an exposition of Malliavin calculus and the notation we use here concerning the stochastic calculus of variations. For G := (G1 ; : : : ; Gm ) 2 (ID1 )m , we denote by G its Malliavin covariance matrix, i.e. the m  m-matrix dened by

Gij :=< DGi ; DGj >L (0;T) : 2

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Vlad BALLY , Denis TALAY

Denition 4.5 We say that the random vector G satises the nondegeneracy assumption

if its Malliavin covariance matrix is a.s. invertible and ?^ G the determinant of the inverse matrix ?G := G?1 satises \ ?^ G 2 Lp ( ) : (40) p1

Our analysis deeply uses the following integration by parts formula (see the section V-9 in Ikeda-Wanabe [6] for the proof of (42) and see [4] for the proof of (43)):

Proposition 4.6 Let G0 2 (ID1)m satisfy the nondegeneracy condition of Denition 4.5

and let G in ID1 . Let fH g be the family of random variables depending on multiindices of length strictly larger than 1 and with coordinates j 2 f1; : : : ; mg, recursively dened in the following way:

Hi (G0 ; G) = H(i) (G0 ; G) m G < D?ijG ; DGj0 >L (0;T) +?ijG < DG; DGj0 >L (0;T) := ? Xn

2

0

j=1

0

+?ijG
G
LGj0 ; H (G0 ; G) = H( ;:::; k ) (G0 ; G) := H k (G0 ; H( ;:::; k? ) (G0 ; G)) :

2

o

0

1

1

1

(41)

Let (
) be a smooth function with polynomial growth. Then, for any multiindex ,

IE [(@ )(G0 )G] = IE [(G0 )H (G0 ; G)] :

(42)

Besides, for any p > 1 and any multiindex , there exist a constant C (p; ) > 0 and integers k(p; ), m(p; ), m0(p; ), N (p; ), N 0 (p; ), such that, for any measurable set A  and any G0 ; G as above, one has

IE [jH (G0 ; G)jp

llA ] p 1

 C (p; ) k?^ G

0

llA kk(p; )

kGkN(p; );m(p; ) kG0kN 0(p; );m0 (p; ) : (43)

In this paper we need the following local version of the preceding, where G0 satises the nondegeneracy condition of Denition 4.5 only locally:

Proposition 4.7 Let G0 2 (ID1)m and let G in ID1.

Suppose that for some multiindex , G is invertible a.s. on the set 0

[G 6= 0]

[

j jjj

[D G 6= 0]

INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 17 and that, for any p  1,



IE ?^ G

0

ll[G6=0]

S

j jjj[D G6=0]

p



0]

] j jjj[D G6=0]

S

j jjj[D G6=0]

! (G0)H (G0; G)

p



 IE ?^ G

0

ll[G>0] S

ll[G6=0] S

j jjj[D G6=0]

j jjj[D G6=0]



p

a:s:

:

Thus, the hypothesis (44) and a uniform integrability argument lead to the conclusion.

4.4 Proof of Proposition 4.1. We only prove (32): the arguments to add in order to get (33) follow the same guidelines as those we used in the proof of Lemma 4.2 in [4]. We also limit ourselves to the process (Xtn ): for (Xt ), the proof below can be simplied, in the sense that a localization procedure is unnecessary (which explains that the result holds for T ? Tn  t  T also). We also suppose df (x) > 0: the case df (x) = 0 only diers by some simplications in the proof. We successively will consider the cases where t is small (less than T2 ) and large (between T2 and T ? Tn ). RR n2675

18

Vlad BALLY , Denis TALAY

4.4.1 Small t: t in [0; T2 ]. Since g(
) is a bounded function, in view of (39) it comes n2 jIEx [g(Xtn)@ u(t; Xtn )]j  k F k1   K(T(T?) t)Q 1 + IExdf (Xtn)2q IEx exp ?c dfT(X?t t) f;L "

q

!#

:

Use the fact that T  T ? t  T2 . Then the inequality (37) permits to conclude.

4.4.2 Large t: t in [ T2 ; T ? Tn ]. Let  2 Cb1(IR) such that (x) = 1 for jxj  14 , (x) = 0 for jxj  21 and 0 < (x) < 1 for jxj 2 ( 41 ; 21 ).

Let t (resp. tn) be the Malliavin covariance matrix of Xt (resp. Xtn), and let ^t (resp.

^tn) be their determinants. Dene n rtn := (^ t ^? ^t ) : t Consider

IEx [g(Xtn )@ u(t; Xtn )] = IEx [(1 ? (rtn ))g(Xtn )@ u(t; Xtn )] + IEx [(rtn )g(Xtn )@ u(t; Xtn )] =: A + B : T 2

Using the inequalities (37) and (39) and then the fact that  T ? Tn , one gets

2 T



1 T ?t

2 jAj  k F k1   K(T(T?) t) exp ?c dTf (?x)t IEx j1 ? (rtn)j2 f;L  n k F k1 Ef;L(T; x) IExj1 ? (rtn )j2 : !

h

h

(46)



n T

since

i1

2

i1

2

Slightly modifying the proof of Lemma 5.1 in [4] to take the boundedness of b(
) and (
) into account, we can prove that for any p > 1, there exists an increasing function K (
) depending on p such that p

[IEx j1 ? (rtn )j2 ]  K (T ) n? ; 80 < t  T : 1 2

4

(47)

It remains to choose p = 4 to get the expected upper bound for A:

jAj k F k1 Ef;L(T; x) :

(48)

Let us now treat B which is the really interesting term. INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 19

Consider Xtn as an element of (ID1 )d . Apply the local Malliavin integration by parts formula (45). Setting

Hn(t) := H (Xtn; g(Xtn )(rtn )) ; it comes

B = IEx [u(t; Xtn )Hn(t)] ; = IEx [(PT ?t f )(Xtn )Hn (t)] : Consider a process (X~t ) which is a weak solution of (1) independent of (Xt ); denote ~ the expectation under by ( ~ ; F~ ; I~P ) the probability space on which (X~t ) is dened, and IE ~IP . It comes: B = IEx Hn(t)I~Ez f (X~T ?t) z=X n : 





t

Now, choose a C 1(IRd ) function with compact support x;f;L (
) such that 2:

p (a) if df (x)  T then z;f;L() = 1 for all ; p (b) if df (x) > T then: (b1) z;f;L() = 0 if df ()  df2(x) , (b2) z;f;L() = 1 if  2 Supp(f ), (b3) for any multiindex j@ yf;L ()j  dfC(x) q  (T)Cq= , where C is uniform with 2

respect to df (x) and where q is positive and depends on the dimension d and on only.

The role of this localization function is to keep the memory of the support of f (
) in the Malliavin integration by parts procedure, in order to make the exponential term of Ef;L (x) appear. Then,

B = =



IEx Hn(t)I~Ez

h



IEx Hn(t)I~Ez

h

f (X~T ?t )x;f;L (X~T ?t )

i



z=Xtn

(@x ;:::;xd F )(X~T ?t )x;f;L (X~T ?t ) 1

i



z=Xtn

:

We now apply the proposition 5.2 in [4]: let X~ (
; !~ ) denote a version of class C 1 of the stochastic ow z ! X~ (z; !~ ); let Y~ (
; !~ ) denote its Jacobian matrix and Z~ (
; !~ ) the inverse matrix of Y~ (
; !~ ); there exists processes (Q~  ) such that (49) Q~  (z )@ f@x ;:::;xd F  X~  (
; !~ )g(z ) a:s: ; (@x ;:::;xd F )(X~ (z )) = X

1

2

jjd

1

At the beginning of the proof we have supposed that df (x) > 0.

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Vlad BALLY , Denis TALAY

and Q~  (z ) is a polynomial function of the coordinates of Z~ (z; !~ ). Thus, choosing z = Xtn and  = T ? t, one gets

B=

X

jjd





~ IEx Hn(t)@z fF  X~T ?t(z; !~ )gQ~ T ?t (z )x;f;L (X~T ?t (z; !~ )) n : IE z=X

t

As in Section 5.1.2 of [4] we observe that Hn (t) is a sum of terms, each one being a product which includes a partial derivative of (
) evaluated at point rtn; we thus may again apply the integration by parts formula (45) with G0 = Xtn; we obtain for some n (t)), processes (H~ ;

B=

X



from which





n (t)F  X~ (z; !~ ) ; I~E IEx H~ ; T ?t z=X n

jBj k F k1

X



t

n (t) : I~E IEx H~ ;



n (t) is a sum of terms, each one being a product which includes a Observe that H~ ; partial derivative of x;f;L (
) evaluated at a point X~T ?t(z ) z=X n and of a partial derivative t of (
) evaluated at point rtn . Thus,

n (t) n (t) = H ~ ; H~ ;

ll[0;1=2](jrtn j) llSupp(x;f;L)





On jrtnj  12 one has 32 ^t  ^tn  21 ^t and therefore 

n (t) n (t) = H ~ ; H~ ;

ll[ 32 ^t  ^tn 21 ^t ] llSupp(x;f;L)



X~T ?t(z; !~ ) z=X n(!) : X~T ?t (z; !~ )



t



z=Xtn(!)

:

We x !~ . The inequality (43) (remember Proposition 4.7) leads to the following inequality, where the Sobolev norms are computed w.r.t. IP on :



n (t; !~ )  C ?n(x) IEx H~ ; t





ll[^ tn (x) 1 ^t (x)] llSupp(x;f;L)



2

X~T ?t (z; !~ )



z=Xtn(x)

n (t)Q ~ T ?t (z )x;f;L (X~T ?t (z; !~ )) n kXtn(x)kN;m H; z=Xt (x)







k

N 0 ;m0

(50)

for some integers k; N; m; N 0 ; m0 . We are now going to treat each term of the right side. First, it is clear that



?nt(x)

ll[^ tn(x) 1 ^t (x)]

2

2k

 K (T ) :

Second, let us check that

I~E





llSupp(x;f;L)

X~ T ?t(z; !~ ) z=X n(x)

t





2  K (T ) exp ?c df (Tx) : 2k !

INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 21

Indeed,

I~E





llSupp(x;f;L)

X~ T ?t(z; !~ ) z=X n(x)



t

"

 IP I~P k X~T ?t(z) z=Xtn(x)

? x k df (x)

#

2

 IP I~P k X~T ?t(z) z=Xn(x) ? Xtn(x) k df4(x) "

+IP



t

k Xtn(x) ? x k df4(x)

"

#

#

:

Using (38) again, we get

I~E





llSupp(x;f;L)

X~T ?t(z; !~ ) z=X n(x)



t

d f (x)2 + exp ?c df (x)2  K (T ) exp ?c T T ?t !

"

!#

:

We now use the fact that T  t  T2 . Next, proceeding as in the proof of Lemma 5.1 in [4] and using the additional hypothesis that b(
) and (
) are bounded, we get sup kXtn (x)kN;m < K (t) : n1

Obviously, from the above condition (b3) of the denition of x;f;L (
) there holds (see the detailed arguments in the proof of Lemma 4.4): 2 I~E H n (t)Q~  (z ) (X~ (z; !~ ))  K (T ) exp ?c df (x)



;

T ?t

x;f;L



T ?t

!



Tq

z=Xtn(x) N 0 ;m0

Combining all the preceding remarks, we have got that 2 jBj k F k K (T ) exp ?c df (x) 1

Tq

T

!

T

:

:

In conclusion, the preceding estimate, (46) and (48) prove that the inequality (32) holds for T2  t  T ? Tn .

4.5 Proof of Proposition 4.2. With (
) and rtn dened as in the preceding proof, consider A := IEx f (X~Tn )(1 ? (rTn ?T=n )) ; B  := IEx (u(T; X~Tn ) ? u(T ? T=n; XTn?T=n + Z n))(rTn ?T=n ) ; C  := IEx (u(T ? T=n; XTn?T=n + Z n) ? u(T ? T=n; XTn?T=n))(rTn ?T=n ) ; D := IEx u(T ? T=n; XTn?T=n)(1 ? (rTn ?T=n )) : h

i

i

h

i

h

h

RR n2675

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22

Vlad BALLY , Denis TALAY

Clearly, it is sucient to prove that jAj, jB  j, jC  j, jD j can be bounded from above by the right side of (36). Let us start with A. We again consider the smooth function x;f;L (
) of the preceding proof. We observe that, for a certain set  of multiindices of length smaller than d, 

Z

A = IEx (1 ? (rTn ?T=n )) f (XTn + z )x;f;L (XTn + z )1=n(z )dz 2

= IEx (1 ? (rTn ?T=n )) 4

X

2

Z



3

F (XTn + z )@ (1=n(z )x;f;L (XTn + z ))dz : 5

Remembering that we have supposed n > df2(x) , we deduce that, for some constant C linearly depending on the L2 (IRd)-norm of the @ 0's, for some q > 0 and Q > 0 there holds

jAj  Observe that

v u u t

C k F k IE (1 ? (rn ))2 IP d (X n )  df (x) ? 1 : 1 x f T x T ?T=n q n TQ 2 n q

"

"

#

df (XTn )  df 2(x) ? n1  kXTn ? xk  df 2(x) : #

"

#

We then conclude by applying the inequalities (38) and (47):

jAj 

K (T ) k F k exp ?c df (x)2 : 1 TQ T !

Next, we observe that 2 B  ? Tn2 IEx (T ? T=n; XTn?T=n + Z n)(rTn ?T=n ) h

i

is a sum of terms of the type (28). We then apply the arguments used in the subsection 4.4.2, especially the integration by parts formula (45) with G0 = XTn?T=n(x). Of course, we also use the fact that Z n is independent of the process (Wt ). For the term C  we directly apply (45). For the term D, we use the same arguments as for A.

5 An exponential bound for the local density q The objective of this section is to prove Proposition 2.1. For convenience we recall it.

Proposition 5.1 Assume (H). Let L be such that UL is non void. INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 23

(i) Then there exists a smooth function (t; z;  ) 2 (0; T ]  IRd  UL ! qt (z;  ) such that, for any measurable and bounded function (
) with a compact support included in UL , one has

IEz (Xt ) =

Z

Supp()

( )qt (z;  )d ; 8z 2 IRd :

(51)

(ii) Let L  1, m be arbitrary integers and let ; be multiindices. There exist positive constants , c and there exists an increasing function K (
) such that, for any 0 < t  T, 2 m 8y 2 U ; 8z 2 IRd ; @ @ z @ y q (z; y)  K (T ) exp ?c k z ? y k : (52)

L

@tm



 t

!

(tVL (y))

t

Inequalities of this type are classical when the innitesimal generator of (Xt ) is strongly elliptic: for example, see Friedman [5]. We have not found (8) in the literature under our hypotheses. Observe that nevertheless it is a variation of (6): the roles of z and y are permuted. To prove the result, we rst note that the Fokker-Planck equation permits to only consider the case of spatial derivatives: from now on, we set m = 0. Theorem 2.1 in Bally and Pardoux [1] provides a localized version of Malliavin's absolute continuity criterion and permits to construct a smooth function qt (z;
) for each (t; z ) 2 (0; T ]  IRd. Here we prove the dierentiability with respect to all the variables. Proof. (i) Consider the sequence of open sets  := f ; V ( ) > g B 0; 1

UL





 : Let   (
) be a smooth function with a compact support in UL and such that   ( ) = 1 in UL . Consider the nite measure dIPXt(z) ( )  ( )dz : Let (z;  ) be a smooth function with a compact support in IRd  IRd and let , be L

multiindices. One has: Z

@z @ 







(z;  )  ( )dIPXt (z)() dz =

Z h



Z h

= IE = IE = RR n2675

\

Z

@z @  (z; Xt (z ))  (Xt (z )) dz

2

@ 



IE @  4







i

(z; Xt (z ))@z f  (Xt (z ))g (z; Xt (z ))

X

j jjj

i



dz



(@   ) (Xt (z ))Q (t; z )

3 5

24

Vlad BALLY , Denis TALAY

for some polynomial functions Q (t; z ) of the derivatives of the ow of X t (z ). We now apply Proposition 4.7: for some positive constant C independent of (
) it holds that Z



@z @ 



(z;  )  ( )dIP

Xt (z) ( )dz



 C k kL1(IRd IRd ) :

Thus, the measure   ( )dIPXt (z) ( )dz has a smooth density qt (z;  ) with respect to Lebesgue's measure. Therefore, for any  > 0 and t > 0, for any smooth function  (
) with a compact support in UL , one has Z

IEz  (Xt ) = qt (z;  )  ( )d : For y 2 UL set (y) := supf; y 2 UL g. Now for (z; y) 2 IRd  UL dene

qt(z; y) := qt(y) (z; y) : By construction this function is a smooth function and satises (51). (ii) We now turn our attention to (52). Let y 2 UL . Dene  (
) as before and set t > 0. We consider the case m = jj = 0 and j j > 0: one can treat the other cases with analoguous arguments (for m > 0 one must use the Fokker-Planck equation in addition). As qt(
;
) is a smooth function, Z

 ( )@ y qt(z; y +  )d : @ y qt(z; y) = lim !0  Thus, integrating by parts and denoting by Yt (z ) the matrix @ z Xt (z ) we get

j@ qt(z; y)j = j lim IE [(@ y ;y;z )(Xt (z ) ? y)]j ; !0 where i y ;y;z ( ) :=  0  ?  i=1 d

Y

1

(

!

Dene

AL (y) = BL (y) =

i ? y ? 1 ll[yi?zi>0] + 0  !

!

)

ll[yi?zi0]

:

 ; VL ( ) > 41 VL(y) ;  ; VL ( ) > 21 VL(y) :









Since the application VL (
) is continuous, the closure of BL (y) is included in AL (y) and one may nd a function y;L (
) in C 1(IRd ) taking its value in [0; 1] such that INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 25

(a) y;L(z) = 0 if z 2 AcL(y) or d(z; BL)  VL(y), (b) y;L(z) = 1 if z 2 BL (y), (c) for any multiindex , j@z y;L(z)j  V C(y)q L where C and q are positive and uniform w.r.t y 2 UL and L. Observe that, for all  small enough, for all  2 IRd, @ y ;y;z ( ? y) = @ ;y;z ( ? y)y;L ( ) : Consequently,

IEz [(@ ;y;z )(Xt ? y)] = IEz [(@ ;y;z )(Xt ? y)y;L (Xt )] : We again apply Proposition 4.7: the Malliavin integration by parts formula implies that the right side is equal to IEz [;y;z (Xt ? y)H (Xt ; y;L (Xt ))] ; with

Z

1

;y;z ( ) := : : : 0 Thus, making  tend to 0, we get "

@ y qt (z; y)

Z

d 0

;y;z (1 ; : : : ; d )d1 : : : dd :

= IEz H (Xt ; y;L (Xt ))

d

Yn

i=1

ll[0;+1)(Xti ? yi ) ll[yi?zi>0]

? ll(?1;0](Xti ? yi) ll[yi?zi0] : oi

H (Xt ; y;L (Xt )) is a sum of terms, each one containing y;L (
) or a derivative of y;L (
), functions which vanish on the set AL (y)c , so that H (Xt ; y;L (Xt )) = H (Xt ; y;L (Xt )) llAL (y) (Xt ) whence

j@ y qt(z; y)j



r

h

i

IEz H (Xt ; y;L (Xt ))2 llAL (y) (Xt ) v u u t

IEz

d

Yh

i=1

ll[0;+1)(Xti ? yi ) ll[yi ?zi>0] + ll(?1;0] (Xti ? yi ) ll[yi?zi0]

i

:

Using the inequality (38) and standard computations (see [8] or [4]) one gets

j@ y qt(z; y)j  K (T ) k llAL(y)(Xt(z))?^ t(z) kk RR n2675

X

2

2 k@ y;L(Xt(z))kN;m exp ?c k z ?t y k

!

26

Vlad BALLY , Denis TALAY

for some integers k, N and m, and a nite set  of multiindices. We now use the condition (c) of the denition of y;L (
). To conclude, it remains to apply the inequality (54) below with A = AL (y) using the fact that VL (A) = VL (AL (y))  1 4 VL (y) > 0. We then get (52).

6 An Lp-estimate for the inverse Malliavin covariance matrix The aim of this section is to prove the following

Proposition 6.1 Assume (H).

Fix two arbitrary integers p and L in IN ?f0g. Then there exists a positive constant  and an increasing function KL;p (
) both depending on p and L which satisfy the following: for any Borel set A, if VL(A) := inf V ( ) > 0 (53) 2A L then, for all 0 < t  T , for all z 2 IRd,

k

llA (Xt (z ))^

t (z )?1 kp 

KL;p (t) : (tVL (A))

(54)

Proof. First we show that it is sucient to prove that, for all p  1 there exist strictly positive constants ; Q; Q0 and an increasing0 function KL;p (
) such that, for all z 2 IRd, for all T  t > 0, for all 0 < " < min(t?Q ; tQ ), for all Borel set A for which (53) holds,

IPz [ llA (Xt )j ^t j  " ; Xt 2 A]  (tVKL;p(A(t)))  "p+2 :

(55)

L

Indeed, we would then have

IEz

h

llA (Xt )j ^t?1jp

i

=

1

X

k=0

IEz

h

1

llA (Xt )j ^t?1jp ll[k llA (Xt)j ^t?1 jk+1] ll[Xt 2A]

 1 + (k + 1)pIPz X



k=1 KL;p (t) tQ00



llA (Xt )j ^t j 

1 (k + 1)p X

+ KL;p (t) t VL (A) k1 kp+2 :

1

k



[Xt 2 A]

\

i



Thus, as by denition 0 < VL (A)  1, we would have obtained (54) (with a new function KL;p (
) and a possibly new constant ). Thus, we are going to prove (55). INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 27

We start with some localizations. Under (H), the function VL (
) is uniformly Lipschitz. Thus, there exists a constant L depending only on L, b(
) and (
) such that

d(z; A)  LVL (A) ) VL(z )  VL2(A)

(56)

:= 2L2+ 1

(57)

B" := [k Xt ? Xt?" t k LVL (A)] :

(58)

Set and

A Girsanov transformation and Tschebyche's inequality show that under (H), for any q  1 and for some increasing function KL;q (
) uniform w.r.t. A,

q

IP ( ? B" )  KVL;q((At))"2q :

(59)

L

We introduce another set of localization. Again denote by Yt(z ) the matrix @ z Xt(z ) and denote by Zt(z ) the inverse matrix of Yt(z ). Set

C" :=

"

  k 1 inf k Z t ?" t kk=1

#

:

(60)

An easy algebraic computation shows that inf k Zt?" t  k= kk=1

1

supkk=1 k Yt?" t  k

;

so that for any q  1 and for some increasing function Kq (
) only depending on b(
), (
) and q, 2q IP ( ? C")  IE sup kYt?" t k  Kq (t)" q : (61) "

#

kk=1

Thus, (59), (61) and 0 < VL (A) < 1 reduce the proof of (55) to the proof of

IPz

h

llA (Xt )j ^t j  "

K (t) ; [Xt 2 A] C" B"   L;p  "p+2 : t VL (A) \

\

i

Instead of keeping t , we will use a new matrix t which we now dene. Set

t :=

RR n2675

d

X

Z

t

i=1 0

(Zsi (Xs))(Zs i (Xs )) ds :

(62)

28

Vlad BALLY , Denis TALAY

Observe that, by the variation of constants formula, one has

t = Ytt Yt :

Since, for all q  1, there exists an increasing function Kq (
) such that k (Y^t)?1 kq < Kq (t) ; it is sucient to prove (62) with t instead of t. For any symmetric nonnegative denite d  d matrix M , one has:

det(M )1=d  kinf < M;  > : k=1 Therefore (observe that 0 < "  1 by denition),

^t1=d

 

d

X

i=1 d X i=1

inf kk=1 inf kk=1

Z

0 Z

t   (Zs i (Xs ))(Zs i (Xs)) ds t

t?" t

  (Zsi (Xs ))(Zs i (Xs )) ds :

Observe that, for t ? " t  s  t the inverse matrix of Ys(x), Zs(x), saties:

Zs(x) = Zt?" t (x)Zs?(t?" t) (Xt?" t (x)) ; so that Z t d X 1=d ^t  kinf i=1 k=1 t?" t

<  Zt?" t ; Zs?(t?" t) (y)i (Xs?(t?" t) (y)) >2



y=Xt?" t

ds :

On C, for k k = 1 one has k  Zt?" t k  1. Thus, on C ,

^t1=d

 = =:

d

X

i=1 d X

inf kk1 inf kk=1

i=1 G1=d t;"

:

Z

t

t?" t Z " t 0

< ; Zs?(t?" t) (y)i (Xs?(t?" t) (y)) >2

< ; Zs (y)i (Xs(y)) >2



y=Xt?" t



y=Xt?" t

ds

ds

Therefore, a sucient condition for (62) is

IPz Gt;"  " ; [Xt 2 A] B"  tKVL;p(A(t)) "p+2 : L h

\

i

(63)

Let us now turn our attention to the localization in B" . On B" [Xt 2 A], one has VL(Xt?" t)  VL2(A) : T

INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS  LAW OF THE EULER SCHEME 29

Thus, a new sucient condition for (62) is Z

IPy

d

" X

inf

Z

" t

i=1 kk=1 0

#

< ; Zs i (Xs ) >2 ds  " ll[VL (y) VL2(A) ] dIPXt?" t (z) (y)

 tKVL;p(A(t)) "p+2 : L

We nally observe that the preceding inequality follows from the estimate (64) in the theorem below (with  = tL+1"1=(2L+1),  = tL(L+1) "?(L+1)=(2L+1) and "  min(tL(L+1) ; t?(L+1)(2L+1))), being dened as in (57), which ends the proof. We have just referred to the following theorem (Theorem 2.17 in [8]):

Theorem 6.2 (Kusuoka-Stroock) Assume (H). There exists a positive constant C depending only on b(
) and (
) and for any L  1 there exist positive constants L , CL such that, for any 0 <  < 1, for any   1, IPy

"

d

X

i=1

inf kk=1

Z

=1=(L+1)

0

< ; Zs

i (X

s

) >2 ds 

 L  C exp ?CV (y)(L+2)L L : L L  #





(64)

References [1] V. BALLY and E. PARDOUX. Malliavin calculus for the solutions of parabolic SPDE's. Submitted for publication, 1995. [2] V. BALLY, P. PROTTER, and D. TALAY. The law of the Euler scheme for stochastic dierential equations. Zeitschrift für Angewandte Mathematik und Mechanik, 1995. [3] V. BALLY and D. TALAY. The Euler scheme for stochastic dierential equations: error analysis with Malliavin calculus. Mathematics and Computers in Simulation, 38:3541, 1995. [4] V. BALLY and D. TALAY. The law of the Euler scheme for stochastic dierential equations (I) : convergence rate of the distribution function. Probability Theory and Related Fields, 102, 1995. [5] A. FRIEDMAN. Partial Dierential Equations of Parabolic Type. Prentice Hall, 1964. [6] N. IKEDA and S. WATANABE. Stochastic Dierential Equations and Diusion Processes. North Holland, 1981.

RR n2675

30

Vlad BALLY , Denis TALAY

[7] A. KOHATSU-HIGA. An approximation to the density of a diusion. Submitted for publication, 1995. [8] S. KUSUOKA and D. STROOCK. Applications of the Malliavin Calculus, part II. J. Fac. Sci. Univ. Tokyo, 32:176, 1985. [9] D. NUALART. Malliavin Calculus and Related Topics. Probability and its Applications. Springer-Verlag, 1995. [10] P. PROTTER and D. TALAY. The Euler scheme for Lévy driven stochastic dierential equations. Rapport de Recherche 2621, INRIA, 1995. Submitted for publication. [11] D. TALAY and L. TUBARO. Expansion of the global error for numerical schemes solving stochastic dierential equations. Stochastic Analysis and Applications, 8(4):94 120, 1990.

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