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1、Available online at wwwsciencedirectCOM 黪二 ScienceDi rect Acta Mathematica Scientia 2014，34B(1)：6572 数学物理学报 http：actamswipmaccn QUASISURE CONVERGENCE RATE OF EULER SCHEME F0R SToCHASTIC DIFFERENTIAL EQUATIONS 肌nlian9 H G(黄文亮) School of Management Shanghai University oScience and TechnologyShanghai 2
2、00093,China Department of Mathematic，East China University of Science and Technology，Shanghai 20023China Email：hwlsqt1 63CO?n Xicheng ZHANG l张希：拳1 SchO0f of Mathematics and Statistics，Wuhan University，Wuhan 430072，China Email：XiehengZhanggmaileom Abstract Let Xt(z)be the solution of stochastic diffe
3、rential equations with smooth and bounded derivatives coefficientsLet ( )be the Euler discretization scheme of SDEs with step 2一”In this note，we prove that for any R0 and，y(0，12)， sup l ( ， )一Xt(x， )lR， ( )2一” ， n1， qe 【0，1】，l0IR whereR， ( )is quasieverywhere finite Key words Euler approximation；qua
4、sisure convergence；SDE 2010 MR Subject Classification 60H15 1 Introduction Consider the following stochastic differential equation(SDE)of It6S type d Xt：= r(X t )，d +6( )d where )t_o1is an mdimensional standard Brownian motion defined on the classical Wiener space(Q， ，P)， ，i=1，-一，d，J：1，m)andbi i=1，d
5、)are bounded smooth functions on Rd with bounded derivatives of all ordersThe unique solution is denoted by t( ) The Euler scheme of SDE(11)is defined by ?= +t O(。n )-d +o 6( )ds，nN Received October 30，2012 ACTA MATHEMATICA SCIENTIA Vo134 SerB or recursively, =X n +盯( )( wt )+b(Xro)(ttn) where sn：=
6、，and denotes the integer part of real number n (12) Up to now，there were many papers devoted to the study of the various convergence for Eulers scheme(see【2，3，6，14etc)It is interesting that the Euler approximation even can be used to construct the solutions for SDEs with discontinuous coefficients i
7、n Gy6ngyKrylov _31On the other hand，for the aim of numerical calculations，in1j Bally and Talay studied the convergence rate of the distribution function for Euler scheme As we known，in the classical probability theory，one can ignore a null set in the sense of probability measureHowever，a more delica
8、te analysis in potential theory shows that the zero probability set can not always be ignoredSince Malliavin7created the stochastic calculus of variation in 1976the analysis over infinite dimensional spaces was developed extensively Meanwhile，in the paper8，Malliavin also initiated the quasisure anal
9、ysis，which is finer than almostsure analysis in probability theoryMore introduction about the quasisure analysis can be found in Malliavin9Ren13and HuangYan4_4 Basing on this consideration，in this paper we mainly prove that Theorem 11 FOr any R0 and，y(0，百1)，there are slim set N and a quasi everywher
10、e finite random variableR， such that for all N。， sup I ( ， )一 ( ， )l R， ( )2一 1， n1 tEO，1l l【R The proof of this theorem is based on a Doobs inequality in terms of(P， )一capacity established in Ren12together with some necessary estimatesAfter some preliminaries in Section 2，we shall prove this result
11、 in Section 3Throughout the paper，C with or without indexes will denote different constantswhose v3，lues are not important 2 Preliminaries We will work on the canonical probability space(Q， ，P；衄)，where Q is the space of continuous functions on0，1starting at zero，and endowed with the topology of the
12、uniform convergenceP the standard Wiener measure the completion of the Borel ofield of Q with respect to P，皿the Cameron-Martin subspace，i e it consists of functions h：0，1】_ which are absolutely continuous and whose derivative h belongs to L (0，1)；is then a Hilbert space with the inner product For hH
13、，let W(h)：一 h(t)dWtLet us first recall some elementary facts about the Malliavin calculus(cf【4，9)Let c be the smooth functional space defined as follows， c：=F：f(W(h1)，W(h )：厂c (R ， )， 皿，1 n；礼N) For FC and h，one defines the gradient operator (J) )=Oif(W(h )，W(h )(，) No1 wLHuang&XCZhang：QUASISURE CONV
14、ERGENCE RATE OF EULER SCHEME 67 The higher derivatives can be defined similarlyNotice that for ca(R)，the following Leibnitzformula holds D (F)= (F)DF For any P1 and kN，the Sobolev space Dp， is defined as the closure ofC with respect to the norm p，k p+lD FIIL (n；皿 ) which by Meyers inequality，is equi
15、valent to(cf9) Yll；， = 一 ) FII where L is the Ornstein-Uhlenbeck operator Given an open set O in Qits capacity is defined as ，k(o) infllFIIp， ：FDp,k,F0，F1 a_s_。n 0) For any subset A C the capacity of A is defined as cp，k(A) infG， (0)，o is an。pen set and c O) IfCp， (A) 0 for allP1 and N，then A is called a slim setIf some property holds except on a slim set，then we say that it holds quasisurely(qs)or quasieverywhere(abbreviated as qe)It is well known that for any element FD。：=np，kDp， ，we can find a redefinition F (unique qs)such that(cf9)：F=F a s and for each pair(P， )and each E0，there exist
