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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

The Law of the Euler Scheme for Stochastic Differential Equations : I. Convergence Rate of the Distribution Function Vlad BALLY , Denis TALAY

N˚ 2244 Mars 1994

PROGRAMME 6 Calcul scientifique, mode´lisation et logiciel nume´rique

ISSN 0249-6399

apport de recherche

1994

The Law of the Euler Scheme for Stochastic Dierential Equations : I. Convergence Rate of the Distribution Function Vlad BALLY  , Denis TALAY  Programme 6 | Calcul scienti que, modelisation et logiciel numerique Projet Omega Rapport de recherche n2244 | Mars 1994 | 18 pages

Abstract:

We study the approximation problem of IE f (XT ) by IE f (XTn), where (Xt) is the solution of a stochastic dierential equation, (Xtn) is de ned by the Euler discretization scheme with step Tn , and f is a given function. For smooth f 's, Talay and Tubaro have shown that the error IE f (XT ) ? f (XTn ) can be expanded in powers of n1 , which permits to construct Romberg extrapolation procedures to accelerate the convergence rate. Here, we prove that the expansion exists also when f is only supposed measurable and bounded, under an additional nondegeneracy condition of Hormander type for the in nitesimal generator of (Xt): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law of XTn and compare it to the density of the law of XT . AMS(MOS) classi cation: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.

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

Universite du Maine, et Laboratoire de Probabilites, Universite Paris 6, 4 Place Jussieu, 75252 Paris Cedex 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 schema d'Euler pour les equations dierentielles stochastiques: I. Vitesse de convergence de la fonction de repartition Resume :

On etudie le probleme de l'approximation de IE f (XT ) par IE f (XTn), ou (Xtn) est de ni par le schema d'Euler de pas Tn , et f est une fonction donnee. Pour des fonctions f regulieres, Talay et Tubaro ont montre que l'erreur de discretisation IE f (XT ) ? f (XTn) peut ^etre developpee en puissances de n1 , ce qui permet de construire des algorithmes d'extrapolation de Romberg pour accelerer la convergence. Dans cet article, on montre qu'un tel developpement existe aussi quand f est supposee seulement mesurable et bornee, sous une condition supplementaire de non degenerescence de type Hormander pour le generateur in nitesimal de (Xt): pour obtenir ce resultat, on utilise les outils du calcul des variations stochastiques. Dans la seconde partie de ce travail, nous considererons la loi de la densite de XTn et la comparerons a la densite de la loi de XT . Classi cation AMS(MOS): 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.

Mots-cle : Equations dierentielles 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 dXt = b(Xt)dt + (Xt)dWt ; (1.1) where (Wt) is a r-dimensional Brownian motion. The problem of computing the expectation Ef (Xt ) on a time interval [0; T ] by a Monte{Carlo algorithm, (Xt) being a diusion process, arises from various motivations. For example, in Random Mechanics, a random dynamical system with a white noise being given, one wants to get the two rst moments of the response of the system, or the probability that the response reaches a certain level. In numerical analysis, this permits to solve parabolic or elliptic Partial Dierential Equations in situations where deterministic algorithms become dicult to use or unecient, especially when the dimension of the state space is large, when the underlying dierential operator is degenerate, or when the objective is to compute the solution only at a few points. In economy, this permits to compute option prices based upon a large panel of assets. The algorithm consists in approximating the unknown process (Xt) by an approximate process (depending on a parameter denoted by n) Xtn which can be simulated on a computer, and in simulating a large number M of independent trajectories of Xtn , so that Ef (Xt) is approximated by: 1 M f (X n (! )) : t i M X

i=1

The resulting error of the algorithm depends on the choice of the approximate process and the two parameters M and n. The eects of M can be described by the Central{Limit Theorem or large deviation results; in practice, one estimates the maximum value of the variance of f (Xt) for t in [0; T ], and then chooses M according to the desired accuracy and the power of the available computer. A sophisticated variance reduction technique has been developed and analysed by Nigel Newton in [8]. Generally M must be large,and, as just mentioned, one chooses a probabilistic technique because the problem is degenerate or high dimensional: therefore, one takes advantages of simple procedures to approximate (Xt). A natural mean is to use a time discretization scheme of the stochastic dierential equation whose (Xt) is the solution: T=n represents the discretization step. For example, the Euler scheme is de ned by n + b(X n ) T +  (X n )(W (1.2) X(np+1)T=n = XpT=n (p+1)T=n ? WpT=n) : pT=n pT=n n T n For pTn  t < (p+1) n , Xt is de ned by n + b(X n ) t ? pT +  (X n )(W ? W Xtn = XpT=n t pT=n) : pT=n pT=n n 

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Vlad BALLY , Denis TALAY

The eects of n can be measured by the quantity: jIE f (XT ) ? IE f (XTn )j : (1.3) Milshtein [6] was the rst to show that the schemes built for the quadratic mean convergence, and L2 estimates of the corresponding errors, are not relevant in that context, since the objective is to approximate the law of (Xt). Talay ([12] and [13]) and, independently, Milshtein [7], have introduced the appropriate methodology to analyse the error (1.3): it consists in writing this dierence as a sum of terms involving the solution of a parabolic PDE (this technique will be used also below). These references provide schemes such that, under smoothness conditions on b, , f : jIE f (XT ) ? IE f (XTn)j  nC ;  = 1; 2 : Several other schemes have then been proposed by Kloeden and Platen [4]. In Talay and Tubaro [14], a more precise result is proven : under the same conditions, the errors corresponding to these schemes can be expanded in terms of powers of n1 , and formulae for the coecients of the expansion can be derived. In Protter and Talay [11], the same result is shown for the Euler scheme applied to stochastic dierential equations driven by general (discontinuous) Levy processes. Here, we will focus our analysis on the simplest scheme, the Euler scheme: as a consequence of the existence of the expansion, linear combinations of results obtained with this scheme and dierent step-sizes permit to reach any desired convergence rate (Romberg extrapolation technique: see Talay and Tubaro [14]). Our objective is to show the existence of the expansion under a much weaker hypothesis on f than in [14]: we will suppose it measurable and bounded (the boundedness could be relaxed); for example, f can be the indicatrix function of a domain: our result applies when one wants to compute probabilities of the type IP [jXT j > K ]. In counterpart, we suppose a nondegeneracy condition which in particular ensures that, for any t > 0 and any x 2 IRd, the law of the random variable Xt(x) has a smooth density with respect to the Lebesgue measure (essentially, this condition is the Hormander condition for the in nitesimal generator of the process): that condition is less restrictive than the uniform ellipticity of the generator, and therefore our result applies for dynamical systems whose solution, representing a pair (position, velocity), cannot have a uniformly elliptic generator. The organization of the paper is the following : in the section 2, we recall some results of the Malliavin calculus that we will use in the sequel, in particular an estimate due to Kusuoka and Stroock on the derivatives of the density of Xt(x); in the section 3, we state and comment our main result; the section 4 is devoted to the proof, except two technical lemmas which are proved in the section 5; in the section 6, we give some extensions of the result. In the second part of this work [2], we will consider the density of the law of XTn and compare it to the density of the law of XT . INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS { LAW OF THE EULER SCHEME 5

Notation. In all the paper,  being a smooth function, the notation @x(t; x; y) means

that the multiindex  concerns the derivation with respect to the coordinates of x, the variables t and y being xed. 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 (@iVj )ij . Finally, we will use the same notation K (
), q, Q, for dierent functions and postive 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 de ned below (see the hypothesis (HU)).

2. Some basic results of the Malliavin calculus One can now nd several expositions of the Malliavin calculus: see, for example, Nualart [9] (we use the notation of this book in preparation) and Ikeda-Watanabe [3]; a short presentation on the applications to the existence of a density for the law of a diusion process can be found in Pardoux [10]. We only introduce the material necessary to our computations. We x a ltered probability space ( ; F ; (Ft); IP ), and a r-dimensional Brownian motion (Wt) on that space. For h(
) 2 L2(IR+; IRr), we denote by W (h) the quantity 0T < h(t); dWt >. Let S be the space of \simple " functionals of the Wiener process W , i.e. the sub-space of L2( ; F ; IP ) of random variables F which can be written under the form F = f (W (h1); : : :; W (hn)) for some n, polynomial function f (
) , hi(
) 2 L2(IR+; IRr ). For F 2 S , we denote by (DtF ) the IRr-dimensional process de ned by n @f (W (h1); : : :; W (hn ))hi(t) : DtF = @x i=1 i R

X

The operator D is closable as an operator from Lp( ) to Lp( ; L2(0; T )), for any p  1. Its domain is denoted by ID1;p, and we de ne the norm

kF k1;p := IE jF jp + kDF kpLp( ;L (0;T )) 1=p : h

i

2

The j -th component of DtF will be denoted by Dtj F . RR n2244

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Vlad BALLY , Denis TALAY

One also de nes the k-th order derivative as the the random vector on [0; T ]k 

whose coordinates are de ned by ;:::;jk F := Djk : : :Dj F ; Dtj ;:::;t tk t k and we denote by IDN;p the completion of S with respect to the norm 1 1

kF kN;p :=

1 1

#1=p N X p p k IE jF j + IE kD F kL2((0;T )k) k=1

"

:

ID1 will denote the space p1 j1 IDj;p. For F 2 S , one also de nes the Ornstein-Uhlenbeck operator L by n n @f @ 2f (W (h ); : : : ; W (h )) < h ; h > ; (W (h1); : : : ; W (hn))W (hi) ? LF = @x 1 n i j i;j =1 @xi@xj i=1 i which is a closable operator. The domain of L includes ID1 . For F := (F 1; : : : ; F m) 2 (ID1 )m, we denote by F the Malliavin covariance matrix associated to F , i.e. the m  m-matrix de ned by

Fij :=< DF i; DF j > : T

T

X

X

De nition 2.1. We will say that the random vector F satis es the nondegeneracy as?1 sumption if the matrix F is a.s. invertible, and the inverse matrix ?F := F satis es k?F k 2 Lp( ) : \

p1

Remark 2.2. The above condition can also be written (we recall that ^F denotes the determinant of F ): 1 2 \ Lp( ) :

^F p1

Our main ingredient is the following integration by parts formula (cf. the section V-9 in Ikeda-Wanabe [3]): Proposition 2.3. Let F 2 (ID1)m satisfy the nondegeneracy condition 2.1, let g be a smooth function with polynomial growth, 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 de ned in the following way: Hi (F; G) = H(i)(F; G) m G < D?ijF ; DF j > +?ijF < DG; DF j > +?ijF
G
LF j ; := ? Xn

H (F; G) = :=

j =1 H( 1 ;:::; k )(F; G) H k (F; H( 1;:::; k?1)(F; G)) :

o

(2.1) INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS { LAW OF THE EULER SCHEME 7 Then, for any multiindex ,

IE [(@g)(F )G] = IE [g(F )H(F; G)] :

(2.2)

We can get the following estimate:

Proposition 2.4. For any p > 1 and any multiindex , there exists a 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 F; G as above, one has

IE [jH (F; G)jp l A] p  C (p; ) k?F l A kk(p; ) kGkN (p; );m(p; ) kF kN 0(p; );m0(p; ) : (2.3) 1

Proof. We apply the Meyer inequality on kLF kp (see the theorem 8.4 of the chapter 5 in Ikeda-Watanabe [3], with k = 2, taking into account the de nition 8.2 in the same chapter):

and the equality

kLF kp  C kF k2;p ;

D?ij = ?

X

k;l

?ik ?jl D kl ;

the result readily follows from the de nition (2.1). We now state another classical result, which concerns the solutions of stochastic differential equations considered as functionals of the driving Wiener process. [A; A0] will denote the Lie brackett of two vector elds A and A0.

De nition 2.5. Let us denote by A0; A1; : : : ; Ar the vector elds de ned by A0(x) = Aj (x) =

d

X

i=1 d X i=1

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

For a multiindex  = (1; : : :; k ) 2 f0; 1; : : : rgk , de ne the vector elds Ai (1  i  r) by induction: A;i = Ai and, for 0  j  r, A(i;j) := [Aj ; Ai]. The Hormander condition is said to hold at the point x if the vector space spanned by all the vector elds Ai, 1  i  r and  multiindex, at the point x, is IRd .

Theorem 2.6. Assume that the coecients b and  are in nitely dierentiable, with

bounded derivatives of order strictly larger than 1. Then, for all x, all t > 0 and i = 1; : : : ; d, Xti (x) belongs to ID1.

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Vlad BALLY , Denis TALAY

Besides, suppose that the Hormander condition holds at some point x. Let t (x) denote the Malliavin covariance matrix corresponding to Xt (x), and ?t (x) its inverse. Then, for any t > 0, one has \ k?t (x)k 2 Lp( ) ; p1

and the random vector Xt (x) has an in nitely dierentiable density pt (x;
).

Actually, Kusuoka and Stroock [5] give an exponential bound for pt (x;
) in terms of the following quadratic forms:

VL (x; ) :=

r

X

X

i=1 jjL?1

< Ai(x);  >2 :

Set

VL (x) = 1 ^ kinf V (x; ) : (2.4) k=1 L The exponential bounds require some smoothness conditions on b and , and are valid for x in the set fx 2 IRd : VL(x) > 0g; as we will apply this estimate for x = Xtn , we assume

(UH) CL := inf x2IR VL (x) > 0 for some integer L. (C) The functions b and  are C 1 functions, whose derivatives of any order are bounded d

(but they are not supposed to be bounded themselves).

Under the conditions (C) and (UH), and a corresponding L being xed (the smallest one, for example), the corollary 3.25 in Kusuoka and Stroock asserts: for any integers m; k and any multiindices  and such that 2m + jj + j j  k, there exist an integer M (k; L), a non decreasing function K (T ) and real numbers C; q; 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: Q kx?yk^ j@tm@x@y pt(x; y)j  tKq (1(T+)(1ky+?kxxkk2))k e?C t kxk ; 80 < t  T : (2.5) (

(1+

1)2 )2

The same theorem provides the following estimate for ?t (x): for any p  1, for some constants C;  one has  (2.6) k?t(x)kp  C 1 +tdLkxk : Remark 2.7. The rate of degeneracy of pt (x; y) as t ! 0 is controlled by that of ?t (x); (2.6) gives an upper bound of order 1=tdL; the lower bound 1=tL?1 is proved in the theorem 5.1 in Bally [1]. 1

The constant 0 of the statement of Kusuoka and Stroock is equal to 1 under (C).

INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS { LAW OF THE EULER SCHEME 9

3. Our main result We denote by L the second-order dierential operator d d L := bi(x)@i + 21 ()ij (x)@ij : i;j =1 i=1 X

X

(3.1)

Consider a measurable bounded function f and u(t; x) := IE f (XT ?t(x)) which solves: @u + Lu = 0 ; @t (3.2) u(T;
) = f (
) : 8 < :

Denote by a the matrix . Let (t; x) be de ned by d d bi(x)ajk (x)@ijk u(t; x) (t; x) = 21 bi(x)bj (x)@ij u(t; x) + 12 i;j =1 i;j;k=1 d @ 2 u(t; x) + 81 aij (x)akl(x)@ijklu(t; x) + 12 @t 2 i;j;k;l=1 d @ @ u(t; x) + 1 d aij (x) @ @ u(t; x) : + bi(x) @t i 2 i;j=1 @t ij i=1 X

X

X

X

X

(3.3)

Theorem 3.1. Let f be a measurable and bounded funtion. Under the hypotheses (UH) and (C), the Euler scheme error satis es

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

(3.4)

the terms Cf (T; x) := 0T IE (s; Xs(x))ds and Qn(f; T; x) have the following property: there exists an integer m, a non decreasing function K (T ) depending on the coordinates of a and b and their derivatives up to the order m, and positive real numbers q; Q such that Q jCf (T; x)j + supn jQn(f; T; x)j  K (T )kf k1 1 + kqxk : R

T The expansion (3.4) was rst obtained by Talay and Tubaro in [14]. No nondegeneracy assumption of Hormander type was necessary, but in counterpart the function f (
) was supposed smooth enough; in that context, one obtains a bound of type jCf (T; x)j + supnjQn(f; T; x)j  K (T ) k@xf k1 : X

jj6

Besides, when f is smooth, the analysis shows that the simulation of Brownian paths is unnecessary to get the existence of the expansion: the algorithm may involve appropriate discrete lawed random variables as well (see [14]). In our context, this property does not remain true. This has no practical incidence: the simulation of the increments of the Wiener process can eciently be performed. RR n2244

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Vlad BALLY , Denis TALAY

4. Proof of Theorem 3.1 The proof of the preceding theorem is based upon the two following technical lemmas, which are interesting by themselves.

Lemma 4.1. Let the function u be de ned by (3.2). Then, for any multiindex , for any

smooth function with polynomial growth g , there exists a non decreasing function K (t) and positive constants q; Q, uniform with respect to n and T , such that

8t 2 [0; T ] ; jIE [g(Xt(x))@u(t; Xt(x))]j  K (T )kf k1 1 + tkqxk : Q

and

8t 2 0; T ? Tn ; jIE [g(Xtn(x))@u(t; Xtn(x))]j  K (T )kf k1 1 + tkqxk : 

Q



(4.1) (4.2)

Lemma 4.2. Under the hypotheses of the Theorem, for some integer q and some non decreasing function K (t), one has that

jIE f (XTn(x)) ? IE (PT=nf )(XTn?T=n(x))j  Kn(2T ) kf k1 (1 + kxkQ) :

(4.3)

For a while, we admit the lemmas (4.1,4.2) which will be proven in the section 5. We start with an easy other lemma.

Lemma 4.3. It holds that IE f (XTn(x)) ? IE f (XT (x)) = 1 kT ; X n (x) I E n2 n kT=n X

!

kn?2 +IE f (XTn(x)) ? IE (PT=nf )(XTn?T=n(x)) +

X

kn?1

Ikn ;

(4.4)

where Ikn can be explicited under a sum of terms, each of them being of the form

IE

Z (k +1)T=n Z s Z s 1 2 \ n (XkT=n(x)) (] (X n (x))@u(s3; Xsn3 (x)) kT=n kT=n kT=n  s3 i +[(Xs3 (x))@u(s3; Xs3 (x)))ds3ds2ds1 ;

h

(4.5)

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

INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS { LAW OF THE EULER SCHEME 11 Proof. We follow [14], just changing the presentation. For a xed z in IRd, we de ne the dierential operator Lz by d d X X Lz g(
) := bi(z)@ig(
) + 21 aij (z)@ij g(
) : i;j =1 i=1 n We note that, for z = XkT=n, Lz is the in nitesimal generator of the diusion process  T . Xtn; kTn  t < (k+1) n As u(t;
) = PT ?tf (
) = IE f (XT ?t(
)), one has X kn IE f (XTn(x)) ? IE f (XT (x)) = IE u(T; XTn(x)) ? u(0; x) =

with

kn?1

n (x) := IE u (k + 1)T ; X(nk+1)T=n(x) ? u kT ; XkT=n n n The It^o formula implies "

kn

kn

= IE

!

Z (k +1)T

n

kT n

kn = IE

n

kT n

x)

kT=n (





kT ; X n (x) ? L u z n kT=n kT=n (x)

It := Lz u(t; Xtn(x)) z=X n

x)

kT=n (

!

:

(4.6)

dt ;

!

?Lu(t; Xtn(x)) + Lz u(t; Xtn(x)) z=X n

Denote and

!



@tu(t; Xtn(x)) + Lz u(t; Xtn (x)) z=X n

from which, using (3.2), one gets Z (k +1)T

!#

dt :



n z=XkT=n (x)

n Jt := Lz u kT ? Lu(t; Xtn(x)) n (x) n ; XkT=n(x) z=XkT=n n n = Lu kT n ; XkT=n(x) ? Lu(t; Xt (x)) : !



!

We have:

kn

= IE

Z (k +1)T

n

kT n

(Jt + It)dt :

We consider It and Jt as smooth functions of the process (Xtn ) and recursively apply the It^o formula, using the fact that the function u solves (3.2), so that Lu solves a similar PDE. We can deduce the following corollary (note that this upper bound in terms of kf k1 was stated nowhere else in the literature before, even for smooth functions f (to our knowledge); but we do not focus on it, since our objective is a stronger statement): RR n2244

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Vlad BALLY , Denis TALAY

Corollary 4.4. There exist a non decreasing function K (T ) and constants q; Q such that: jIE f (X n(x)) ? IE f (XT (x))j  K (T )kf k1 (1 + kxkQ) 1 : (4.7) T

Tq n Proof. We apply the lemma 4.2, and for k < n ? 1, we apply the estimates of the lemma 4.1 n (x)) and I n . to ( kTn ; XkT=n k Let us now end our proof. The expansion (4.4) can be written again under the following form:

IE f (XTn(x)) ? IE f (XT (x)) = 1 IE kT ; X (x) + 1 IE kT ; X n (x) ? kT ; X (x) n2 kn n kT=n n2 kn n kT=n n kT=n !

X

+

X

kn?1

X

!

"

!#

Ikn :

Applying the preceding corollary, we get that, for any k such that nk  21 , one has

K (T ) kf k (1 + kxkQ) : n (x) ? IE kT ; X IE kT ; X kT=n (x)  kT=n n n nT q 1 !

!

For nk  12 , one applies the expansion (4.4) to the function fn;k (x) := ( kTn ; x). Denote un;k (t; x) := PT ?tfn;k (x) and n;k (t; x) the function de ned in (3.3) with un;k instead of u; then one has 1 kT ; X n (x) + n (x) ? IE kT ; X IE kT ; X Jnj ; kT=n (x) = 2 IE n;k kT=n n n n n kT=n j k?1 !

!

!

X

where the Jnj 's are sums of terms of type (4.5) with un;k instead of u. Therefore, in order to prove (3.4), we are going to check that for g 2 Cb1(IRd) and s  kTn  T2 , one has

jIE [g(Xsn(x))@xun;k (s; Xsn(x))]j  K (T )kf k1 1 +Tkqxk : Q

(4.8)

Remembering the de nition of and n;k , we note that

@xun;k (s; z)

= = =

Z

@xpT ?s (z; y) (kT=n; y)dy

X

jj6 X

jj6

Z

Z

@xpT ?s (z; y)h(y)@y u(kT=n; y)dy @xpT ?s (z; y)h(y)

Z

@y pT ?kT=n(y; y0)f (y0)dy0dy ; INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS { LAW OF THE EULER SCHEME 13

where the h's are functions which depend on the coordinates of a and b. Since T ? s  T ? kT=n > T=2, one may apply (2.5) to get j@xun;k (s; z)j  KT(Tq ) (1 + kxkQ)kf k1 ; which readily implies (4.8). Now, using It^o's formula and the estimate (4.1), we get 1 kT ; X (x) ? T IE (s; X (x)) ds  K (T ) kf k (1 + kxkQ) : I E s 1 n kn?1 n kT=n n 0

!

X



Z

We note that (4.1) also ensures that 0T IE j (s; Xs (x)) jds is nite. That ends the proof of the theorem 3.1. R

5. Proof of Lemmas 4.1 and 4.2 We rst state a technical lemma.

Lemma 5.1. Under the above hypotheses, for any p > 1 and j  1, there exist an integer

Q and a non decreasing function K (t) such that sup kXtn (x)kj;p < K (t)(1 + kxkQ) n1

and

sup kXt(x) ? Xtn(x)kj;p < Kp(nt) (1 + kxkQ) : n1

Proof. We just have to mimic the classical computations giving estimates for kXt (x)kj;p. For example, let us show how we can proceed for j = 1. Let n (t) denote pTn , where the T T k n integer p is such that pTn  t < (p+1) n . We remark that, for t ?  > n , D Xt satis es:

Dk Xtn

=

r Zt X n @l(Xnn(s))Dk Xnn (s)dWs (Xn()) + n ( )+T=n  l=1 Z t @b(Xnn(s))Dk Xnn (s)ds : + n  ()+T=n

Under (C), a classical use of Gronwall's lemma permits to get sup sup kXtn (x)k1;p < K (T )(1 + kxkQ): n1 0tT

For other values of j and for the dierence Xt(x) ? Xtn (x), we proceed in the same way.

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5.1. Proof of Lemma 4.1. We only provex the part (4.2), the part (4.1) being

treated with the same arguments. We recall that @(t; x; y) means that the multiindex  concerns the derivation with respect to the coordinates of x, the variables t and y being xed. As u(t; x) = PT ?t(f )(x), one has

@xu(t; x) =

Z

@xpT ?t(x; y)f (y)dy :

Now we use the estimate (2.5) and get j@xpT ?t(x; y)j  K(T(T??t)tq) (1 + kxkQ) (x; y) (1 + ky1? xk2)k ; so that j@xu(t; x)j  K (T ) (Tkf?k1t)q (1 + kxkQ) :

(5.1)

5.1.1. Small t. We rst consider the case 0  t  T2 . As T ? t  T=2  t, the

inequality (5.1) yields (4.2).

5.1.2. Large t. Now, let us treat the situation T2  t  T ? Tn . The preceding argument cannot be used, since for  ! T the measure pT ? (x; y)dy converges weakly to the Dirac measure at point x. The principle of the rest of the proof is the following: in order to get rid of the derivatives of u(t; x), we will use Malliavin's integration by parts formula with respect to the functional Xtn , which should be non degenerated for t  T=2; estimates on k?nt kp can be directly obtained under a uniform ellipticity condition but, under (UH) only, we are led to compare k?nt kp and k?t kp and we will use a localization argument:

 let 0 be the set of events where j ^tn ? ^tj is larger than C ^t; to prove that IP ( 0) is

small, we will use two facts: rst, (Xtn ) is a \good" approximation of (Xt); second, the k ^t?1kp's are nite;  on the complementary set of 0, j ^tn ? ^tj is small, which (roughly speaking) means that the Malliavin covariance matrix of Xtn behaves like that of Xt, and allows integrations by parts of type 2.2 with a good control of the Lp-norms of the variables H .

Let  2 Cb1 (IR) such that (x) = 1 for jxj  41 , (x) = 0 for jxj  12 and 0 < (x) < 1 for jxj 2 ( 41 ; 12 ). Let us introduce n rtn := (^ t ^? ^t) : t

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STOCHASTIC DIFFERENTIAL EQUATIONS { LAW OF THE EULER SCHEME 15

We have

IE [g(Xtn)@xu(t; Xtn )] = IE [g(Xtn)@xu(t; Xtn)(1 ? (rtn ))] + IE [g(Xtn)@xu(t; Xtn)(rtn )] := A + B : To upper bound A, we use (5.1):

jAj  K (T ) (Tkf?k1t)q IE j1 ? (rtn)j :

(5.2)

Since 1 ? (rtn ) = 0 for jrtnj  41 , one has

IE j1 ? (rtn )j

1 = IP j ^n ? ^ j  ^t  IP t t 4 4  IP ^t  n11=4 + IP j ^tn ? ^tj  4n11=4 = IP ^t?1  n1=4 + IP j ^tn ? ^tj  4n11=4 : 



h

jrtn j 

"





#



i







Thus, for any p  1, one has

IE j1 ? (rtn )j  n?p=4IE j ^tj?p + (4n1=4)pIE j ^tn ? ^tjp : But (see the lemma 5.1): kXt ? Xtn k1;p  K (t)(1 + kxkQ)n?1=2, so that

k ^tn ? ^t kp  (1 + kxkQ)K (t)n?1=2 : On the other hand, under our nondegeneracy assumption, one has (see (2.6))  1 + k x k ? 1 sup k ^t kp < C tdL : T tT 2

As a consequence, it holds that 1 ; IE j1 ? (rtn)j  K (T )(1 + kxkQ)n?p=4 tdL

from which and (5.2), remembering that for t  T ? Tn , (T ? t)q  Cnq , one gets 1 : jAj  kf k1 nq?p=4K (T )(1 + kxkQ) tdL (5.3) To obtain the desired result, it just remains to choose p = 4q.

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We now treat B . We apply the proposition (2.3), and get B = IE [u(t; Xtn)Hn ] where Hn := H(Xtn ; g(Xtn )(rtn )). First, we observe that Hn is a sum of terms, each one being a product which includes a partial derivative of  evaluated at point rtn. From the de nition of  it follows that Hn = Hn l [0;1=2](rtn ). On jrtn j  12 one has ^tn  21 ^t and therefore Hn = Hn l [^ tn ^t] : Consequently, jB j  C kf k1 IE jHn l [^ tn ^t]j : We apply (2.3) and obtain, for some integers k; N; N 0 : jB j  C kf k1 ?nt l [^ tn ^t ] k kXtn (x)kN;m kg(Xtn )(rtn )kN 0;m0 1 2

1 2



1 2



  1 + k x k 1 + k x k  C kf k1 k?tkk tdL  C kf k1 tdL :

5.2. Proof of Lemma 4.2. The preceding proof cannot apply to treat nn?1

(de ned in (4.6)) because the last argument before (5.3) cannot be used. We still localize by introducing , but we do it at once: set n ))j ; a := jIE [(f (XTn(x) ? IE PT=nf (XTn?T=n(x)) (1 ? (rT= 2 n n n b := jIE [(f (XT (x) ? IE PT=nf (XT ?T=n(x)) (rT=2)j : = jIE [u(T; XTn(x)) ? u(T ? T=n; XTn?T=n(x)]j : Clearly

n )j  K (T ) kf k (1 + kxkQ) : a  2kf k1 IE j1 ? (rT= 1 2 n2 We then proceed as in the proof of the lemma 4.3, to decompose kn in It and Jt, and we apply the arguments used in the subsection 5.1.2 to treat the term b.

6. Extensions In the theorem 3.1, the boundedness hypothesis on f can be relaxed: the preceding technique can be improved to treat the case of functions f which are measurable and have a polynomial growth, i.e such that 8x ; kf (x)k  Cf (1 + kxkqf ) INRIA

STOCHASTIC DIFFERENTIAL EQUATIONS { LAW OF THE EULER SCHEME 17

for some Cf and qf ; then in all the estimates of the proof, kf k1 must be replaced by a constant C depending on Cf and qf ; indeed, instead of upper bounding quantities of type kf (Xtn (x))kLp( ) by kf k1, one can use that IE kXtn(x)kp can be upper bounded by C (1 + kxkq), where the constants C and q are uniform in n. One can also show the existence of an expansion up to any order: as in [14], instead of bounding the Ikn's of the expansion (4.4), one can apply the technique used at the end of the proof of the theorem 3.1 and make appear an integral of type n1 0T IE 1 (s; Xs(x)) ds (for some appropriate function 1); this operation makes appear a new remaining term for which one applies the lemmas 4.1 and 4.2. Finally, the result can be extended to other schemes. The numerical bene t is less clear, since they involve derivatives of the coecients and therefore require a larger computational eort than the Euler scheme (they also may lead to larger coecients of nTi in the expansion of the error). R

2

7. Conclusion We have proved that the error corresponding to the approximation of IE f (XT ) by IE f (XTn), XTn being given by the Euler scheme, can be expanded in terms of n1 when f is a bounded and measurable function, under an hypothesis of uniform hypoellipticity. It now remains to give estimates on the convergence rate of the density of XTn to the density of XT (when they do exist). This will done in the second part of this work, in preparation [2].

References [1] V. BALLY. On the connection between the Malliavin covariance matrix and Hormander's condition. J. of Functional Analysis, 96(2):219{255, 1991. [2] V. BALLY, D. TALAY. The law of the Euler scheme for stochastic dierential equations (II) : convergence rate of the density. In preparation, 1994. [3] N. IKEDA, S. WATANABE. Stochastic Dierential Equations and Diusion Processes. North Holland, 1981. [4] P.E. KLOEDEN, E.PLATEN. Numerical Solution of Stochastic Dierential Equations. Springer{Verlag, 1992. [5] S. KUSUOKA, D. STROOCK. Applications of the Malliavin Calculus, Part II. J. Fac. Sci. Univ. Tokyo, 32:1{76, 1985.

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[6] G.N. MILSHTEIN. A method of second-order accuracy integration of stochastic dierential equations. Theory of Probability and Applications, 23, 1976. [7] G.N. MILSHTEIN. Weak approximation of solutions of systems of stochastic dierential equations. Theory of Probability and Applications, 30:750{766, 1985. [8] N.J. NEWTON. Variance reduction for simulated diusions. SIAM Journal of Applied Mathematics, 1994 (to appear). [9] D. NUALART. Malliavin Calculus and Related Topics. Book in preparation. [10] E. PARDOUX. Filtrage non lineaire et equations aux derivees partielles stochastiques associees. In Cours a l'Ecole d'Ete de Probabilites de Saint-Flour XIX, number 1464 in Lecture Notes in Mathematics. Springer-Verlag, 1991. [11] P. PROTTER, D. TALAY. Convergence rate of the Euler scheme for stochastic dierential equations driven by Levy processes. In preparation, 1994. [12] D. TALAY. Ecient numerical schemes for the approximation of expectations of functionals of S.D.E. In J. Szpirglas H. Korezlioglu, G. Mazziotto, editor, Filtering and Control of Random Processes, volume 61 of Lecture Notes in Control and Information Sciences. Proceedings of the ENST-CNET Colloquium, Paris, 1983, SpringerVerlag, 1984. [13] D. TALAY. Discretisation d'une E.D.S. et calcul approche d' esperances de fonctionnelles de la solution. Mathematical Modelling and Numerical Analysis, 20(1), 1986. [14] D. TALAY, 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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