Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem

For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically neg...

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Date:	2008
Main Author:	Kulik, A.M.
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spelling	irk-123456789-1644862020-02-10T01:28:06Z Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem Kulik, A.M. Статті For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain non-trivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times of multidimensional diffusions are given. Для різницєвих наближень багаroвимiрних дифузій доведено локальну граничну теорему зі зрізанням. При дуже слабких умовах на розподіли різницевих членів ця теорема стверджує, що ймовірності переходу таких наближень після видалення певних доданків, якими в асимптотичному сенсі можна знехтувати, мають щільності, які рівномірно прямують до щільності ймовірності переходу граничної дифузії та задовольняють певні рівномірні оцінки дифузійного типу Доведення базується на новому варіанті числення Маллявена для добутку скінченної сім'ї мір, які можуть містити нетривіальні сингулярні компоненти. Наведено застосування до рівномірного оцінювання коефіцієнта перемішування та швидкості збіжності для різницевих наближень стохастичних диференціальних рівнянь та до збіжності різницевих наближень локальних часів багатовимірних дифузій. 2008 Article Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem / A.M. Kulik // Український математичний журнал. — 2008. — Т. 60, № 3. — С. 340–381. — Бібліогр.: 14 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164486 519.21 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Kulik, A.M. Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem Український математичний журнал
description	For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain non-trivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times of multidimensional diffusions are given.
format	Article
author	Kulik, A.M.
author_facet	Kulik, A.M.
author_sort	Kulik, A.M.
title	Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem
title_short	Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem
title_full	Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem
title_fullStr	Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem
title_full_unstemmed	Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem
title_sort	malliavin calculus for difference approximations of multidimensional diffusions: truncated local limit theorem
publisher	Інститут математики НАН України
publishDate	2008
topic_facet	Статті
url	http://dspace.nbuv.gov.ua/handle/123456789/164486
citation_txt	Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem / A.M. Kulik // Український математичний журнал. — 2008. — Т. 60, № 3. — С. 340–381. — Бібліогр.: 14 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT kulikam malliavincalculusfordifferenceapproximationsofmultidimensionaldiffusionstruncatedlocallimittheorem
first_indexed	2025-07-14T17:02:06Z
last_indexed	2025-07-14T17:02:06Z
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fulltext	UDС 519.21 A. M. Kulik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv) MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL DIFFUSIONS: TRUNCATED LOCAL LIMIT THEOREM* ЧИСЛЕННЯ МАЛЛЯВЕНА ДЛЯ РIЗНИЦЕВИХ НАБЛИЖЕНЬ БАГАТОВИМIРНИХ ДИФУЗIЙ: ЛОКАЛЬНА ГРАНИЧНА ТЕОРЕМА ЗI ЗРIЗАННЯМ For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain non-trivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times of multidimensional diffusions are given. Для рiзницевих наближень багатовимiрних дифузiй доведено локальну граничну теорему зi зрi- занням. При дуже слабких умовах на розподiли рiзницевих членiв ця теорема стверджує, що ймовiрностi переходу таких наближень пiсля видалення певних доданкiв, якими в асимптотичному сенсi можна знехтувати, мають щiльностi, якi рiвномiрно прямують до щiльностi ймовiрностi пе- реходу граничної дифузiї та задовольняють певнi рiвномiрнi оцiнки дифузiйного типу. Доведення базується на новому варiантi числення Маллявена для добутку скiнченної сiм’ї мiр, якi можуть мiстити нетривiальнi сингулярнi компоненти. Наведено застосування до рiвномiрного оцiнювання коефiцiєнта перемiшування та швидкостi збiжностi для рiзницевих наближень стохастичних ди- ференцiальних рiвнянь та до збiжностi рiзницевих наближень локальних часiв багатовимiрних дифузiй. Introduction. Consider a diffusion processX in Rd defined by the stochastic differential equation X(t) = X(0) + t∫ 0 a(X(s)) ds+ t∫ 0 b(X(s)) dW (s), t ∈ R+, (0.1) and a sequence of processesXn, n ≥ 1, with their values at the time moments k n , k ∈ N, defined by a difference relations Xn ( k n ) = Xn ( k − 1 n ) + a ( Xn ( k − 1 n )) 1 n + b ( Xn ( k − 1 n )) ξk√ n , (0.2) and, at all the other time moments, defined in a piece-wise linear way: *The research was partially supported by the Ministry of Education and Science of Ukraine, project № GP/F26/0106. c© A. M. KULIK, 2008 340 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 341 Xn(t) = Xn ( k − 1 n ) +(nt−k+1) [ Xn ( k n ) −Xn ( k − 1 n )] , t ∈ [ k − 1 n , k n ) . (0.3) Here and below, W is a Wiener process valued in Rd, {ξk} is a sequence of i.i.d. random vectors in Rd, that belong to the domain of attraction of the normal law, are centered and have the identity for covariance matrix. Under standard assumptions on the coefficients of the equations (0.1), (0.2) (local Lipschitz condition and linear growth condition), the distributions of the processes Xn in C(R+,Rd) with the given initial value Xn(0) = x converge weakly to the distribution of the process X with X(0) = x [1]. Thus, it is natural to call the sequence {Xn} the difference approximation for the diffusion X. Consider the transition probabilities for the processes X, Xn: Px,t(dy) ≡ P ( X(t) ∈ dy ∣∣X(0) = x ) , Pn x,t(dy) ≡ P ( Xn(t) ∈ dy ∣∣Xn(0) = x ) , t > 0, x ∈ Rd. It is well known [2] that if the coefficients a, b are Hölder continuous and bounded and the matrix b · b∗ is uniformly non-degenerate, then Px,t(dy) = pt(x, y) dy. The function{ pt(x, y), t ∈ R+, x, y ∈ Rd } (the transition probability density for X) possesses the estimate pt(x, y) ≤ C(T ) t− d 2 exp ( −γ‖y − x‖2 t ) , t ≤ T, x, y ∈ Rd. (0.4) The general question, that motivates the present paper, is whether any (more or less restrictive) conditions can be imposed on the coefficients a, b and the distribution of ξk in order to provide that Pn x,t(dy) = pn t (x, y) dy for n large enough, the densities pn possess an estimate analogous to (0.4) and pn converge to p in an appropriate way. Such a question both is interesting by itself and has its origin in the numerous applications, such as nonparametric estimation problems in time series analysis and diffusion models (see the discussion in the Introduction to [3]), the uniform bounds for the mixing coefficients of the difference approximations to stochastic differential equations (see [4] and Subsection 4.1 below), the difference approximation for local times of multidimensional diffusions (see [5] and Subsection 4.2 below). In the current paper, we consider the question exposed above in a slightly modified setting. For the distributions Pn, we prove the result that we call the truncated local limit theorem. Let us explain this term. We show that the kernel Pn can be decomposed into the sum Pn = Qn + Rn in such a way that both Qn and Rn are a non-negative kernels and (i) for Qn, its density qn exists, satisfies an analogue of (0.4) and converges to p; (ii) for Rn, its total mass can be estimated explicitly and converges to 0. The kernel Qn represents the main term of the distribution Pn and satisfies the local limit theorem; the kernel Rn represents the remainder term, and typically decreases rapi- dly (see statements (iii) and (iii′) of Theorem 1.1 below). Such kind of a representation appears to be powerful enough to provide non-trivial applications (see Section 2 below). On the other hand, the conditions that we impose on the distribution of ξk in order to provide such a decomposition to exist are very mild; in a simplest cases, these conditi- ons have "if and only if" form (see Theorem 1.2 below). Our main tool in the current research is a certain modification of the Malliavin calculus. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 342 A. M. KULIK Let us make a brief overview of the bibliography in the field. Malliavin calculus have not been used widely for studying the properties of the distributions of the processes defined by the difference relations of the type (0.2), (0.3). The only paper in this direction available to the author is [4], where rather restrictive conditions are imposed both on the coefficients (b should be constant) and the distribution of ξk (it should possess the density from the class Cd). A powerful group of results is presented in the papers [3, 6], where a modification of the parametrix method in a difference set-up is developed. When applied to the problem formulated above, the results of [3, 6] allow one to prove that pn converge to p with the best possible rate O ( 1√ n ) . However, conditions imposed on the distribution of ξk in [3, 6], are somewhat more restrictive than those used in our approach. For instance, condition (A2) of [3] requires, in our settings, ξk to possess the density of the class C4(Rd) (compare with the condition (B3) in Theorem 1.1 below). The paper is organized in the following way. In Section 1, we formulate the main theorem of the paper together with its particular version, that is an intermediate between classic Gnedenko’s and Prokhorov’s local limit theorems. In the same section, we discuss briefly some possible improvements of the main result. In Section 2, two applications are given. In Section 3, the construction of the partial Malliavin calculus, that is our main tool, is explained in details. In Section 4, the proofs of the main results are given. 1. The main results. 1.1. Formulation. Let us introduce the notation. We write ‖ · ‖ for the Euclidean norm, not indicating explicitly the space this norm is written for. The adjoint matrix for the matrix A is denoted by A∗. The classes of functions, that have k continuous derivatives, and functions, that are continuous and bounded together with their k derivatives, are denoted by Ck and Ck b , correspondingly. The derivative (the gradient) is denoted by ∇, the partial derivative w.r.t. the variable xr is denoted by ∂r. The Lebesgue measure on Rd is denoted by λd. For the measure µ on B(Rd), µac denotes its absolutely continuous component w.r.t. λd. Any time the kernel Pn is decomposed into a sum Pn = Qn + Rn, we mean that the kernels Qn, Rn are non- negative; the same convention is used for decompositions of measures, also. In order to simplify notation we consider the processes defined by (0.1), (0.2) and (0.3) for t ∈ [0, 1] only. Of course, all the statements given below have their straightforward analogues on an arbitrary finite time interval [0, T ]. Through all the paper, κ is a fixed integer, κ ≥ 4. We denote ε(κ) = κ2 − 3κ− 2 2κ+ 2 . We also denote, by µ, the distribution of ξ1. Theorem 1.1. Let the following conditions hold true: (B1) a ∈ C(d+2)2 b (Rd,Rd), b ∈ C(d+2)2 b (Rd,Rd×d) and there exists β = β(b) > 0 such that (b(x)b∗(x)v, v)Rd ≥ β‖v‖2, x, v ∈ Rd; (Bκ 2 ) E‖ξ1‖κ < +∞; (B3) there exist α ∈ (0, 1) and bounded open set U ⊂ Rd such that dµac dλd ≥ α λd(U) 1IU λd-a.s. Then Pn can be represented in the form Pn = Qn +Rn in such a way that ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 343 (i) Qn x,t(dy) = qn t (x, y) dy and qn → p, n→ +∞, uniformly on the set [δ, 1]×Rd× × Rd for every δ ∈ (0, 1); (ii) there exist constants B, C, γ > 0 such that, for t ∈ [0, 1], qn t (x, y) ≤  Ct− d 2 exp ( −γ‖x− y‖2 t ) , ‖x− y‖ ≤ tBn 1 κ+1 , Ct− d 2 exp ( −γn 1 κ+1 ‖x− y‖ ) , ‖x− y‖ > tBn 1 κ+1 , in addition, for every p > 1 there exists Cp > 0 such that, for t ∈ [0, 1], x, y ∈ Rd, qn t (x, y) ≤ Cpt − d 2 ( 1 + ‖x− y‖2 t )−p ; (iii) there exist constants D, ρ > 0 such that Rn x,t(Rd) ≤ D [ n−ε(κ) + e−ρnt ] , x ∈ Rd, t ∈ [0, 1]. If the condition (Bκ 2 ) is replaced by the stronger condition (Bexp 2 ) ∃κ > 0 such that Eexp[κ‖ξk‖2] < +∞, then the following stronger analogues of (ii), (iii) hold true: (ii′) there exist constants C, γ > 0 such that qn t (x, y) ≤ Ct− d 2 exp ( −γ‖x− y‖2 t ) , t ∈ [0, 1], x, y ∈ Rd; (iii′) there exist constants D, ρ > 0 such that Rn x,t(Rd) ≤ De−ρnt, x ∈ Rd, t ∈ ∈ [0, 1]. Let us formulate separately a modification of Theorem 1.1 in the most studied partial case is a ≡ 0, b ≡ IRd . In this case, Xn (1) is just the normalized sum n− 1 2 ∑n k=1 ξk and the limiting behavior of the distributions of such kind of a sums is given by the Central Limit Theorem. For the densities of the truncated distributions, the following criterium can be derived. We denote by Pn the distribution of n− 1 2 ∑n k=1 ξk. Theorem 1.2. The following statements are equivalent: 1. There exists n0 ∈ N such that [Pn0 ] ac is not equal to zero measure. 2. There exists a representation of Pn in the form Pn = Qn +Rn, such that (2i) Qn(dy) = qn(y) dy and supy∈Rd ∣∣∣qn(y)− (2π)− d 2 e− ‖y‖2 2 ∣∣∣→ 0, n→∞; (2ii) there exist constants D, ρ > 0 such that Rn(Rd) ≤ De−ρn, n ∈ N. The well known theorem by Prokhorov states that the given above statement 1 is equivalent to L1-convergence of the density of [Pn]ac to the standard normal density (see [7], Theorem 4.4.1 for the case d = 1). There exist examples showing that, even while P1 � λd, the density of Pn may fail to converge to the standard normal density uniformly (see [7], Ch. 4, § 3 for the example by Kolmogorov and Gnedenko). The criterium of the uniform convergence is given by another well known theorem by Gnedenko: such a convergence holds if and only if there exists n0 ∈ N such that Pn0 possesses a bounded density (see [7], Theorem 4.3.1 for the case d = 1). Theorem 1.2 shows the following curious feature: under condition of the Prokhorov’s criterium, some exponentially negligible remainder term can be removed from the total distribution in such a way that, for the truncated distribution, the statement of the Gnedenko’s theorem holds. This feature does not seem to be essentially new; one can provide it by using the ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 344 A. M. KULIK Fourier transform technique, that is the standard tool in the proofs of the Prokhorov’s and Gnedenko’s theorems. We give a simple proof of Theorem 1.2 using the partial Malliavin calculus, developed in Section 3 below. This illustrates that the partial Malliavin calculus is a powerful tool that allow one to provide local limit theorems in a precise (in some cases, an “if and only if ”) form. 1.2. Some possible improvements. In the present paper, in order to keep exposition reasonably short and transparent, we formulate the main results not in their widest possible generality. In this subsection, we discuss shortly what kind of improvements can be made in the context of our research. 1. Difference relation (0.2) is written w.r.t. uniform partitions {0 = t0n < t1n < . . .}, tkn = k n , k ∈ Z+, n ∈ N. Without a significant change of the proofs, one can prove analogues of Theorem 1.1 for the processes, defined by the difference relations of the type (0.2) with ξk√ n replaced by ξk √ tkn − tk−1 n and partitions {tkn} satisfying condition ∃c, C, d, D > 0: lim inf n→+∞ 1 n # { k ∣∣∣tkn ≤ t, (tkn − tk−1 n ) ∈ [ c n , C n ]} ≥ td, lim sup n→+∞ 1 n # { k ∣∣∣tkn ≤ t, (tkn − tk−1 n ) ∈ [ c n , C n ]} ≤ tD, t ∈ (0, 1]. (1.1) 2. One can, without a significant change of the proofs, replace the sequence of i.i.d. random vectors {ξk} in (0.2) by a triangular array {ξn,k, k ≤ n} of independent random vectors, possibly not identically distributed, having zero mean and identity for the covariance matrix. Under such a modification, condition (Bκ 2 ) should be replaced by supn,k E‖ξn,k‖κ < +∞, and condition (B3) by (B′3) ∃α, r > 0, xn ∈ Rd : d[µn,k]ac dλd ≥ α1IB(xn,r) λ d-a.s., here µn,k denotes the distribution of ξn,k, B(x, r) denotes the open ball in Rd with the centrum x and radius r. Also, the phase space for ξn,k may be equal Rm with m ≥ d (note that the case m < d is excluded by the condition (B1)). 3. Under an appropriate regularity conditions on a, b, Malliavin’s representation, analogous to (3.25), can be written for the derivatives of the truncated density of an arbitrary order with respect to both x and y. Thus, after some standard technical steps, one can obtain the following estimate, that generalize statement (ii′) of Theorem 1.1: for a given k, l ∈ N, ∂k+l ∂xk∂yl qn t (x, y) ≤ Ck+lt − d+k+l 2 exp ( −γ‖y − x‖2 t ) under (B1), (Bexp 2 ), (B3) and a ∈ C(d+k+l+1)2 b (Rd,Rd), b ∈ C(d+k+l+1)2 b (Rd,Rd×d). 4. Theorem 3.1 provides the truncated limit theorem without essential restrictions on the structure of the functionals. For instance, one can apply this theorem in order to obtain a truncated local limit theorem for difference approximations of integral functionals, etc. 5. Like the Malliavin calculus for (continuous time) diffusion processes, the partial Malliavin calcucus, developed in Section 3, can be applied when the diffusion matrix is not uniformly elliptic, but locally elliptic, only. However, the changes that should be done in the proof are significant; in particular, Theorem 3.1 is not powerful enough ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 345 to cover this case. Thus we postpone the detailed investigation of this case (and more generally, the case of coefficients satisfying an analogue of Hörmander condition) to some further research. 6. In the present paper, we concentrate on the individual estimates for the densities qn t and do not deal with the convergence rate in the statement (i) of Theorem 1.1. The (seemingly) possible way to establish such a rate is to write the Malliavin’s representati- on, analogous to (3.25), for the limiting density p and then construct both the functionals Xn(t) ≡ fn andX(t) = f and the corresponding weights Υfn and Υf , involved into the Malliavin’s representation, on the same probability space with a controlled L2-distance between (fn,Υfn) and (f,Υf ). Since the question about the estimates in the strong invariance principle for the pair (fn,Υfn) is far from being trivial, we postpone the detailed investigation of the rate of convergence in Theorem 1.1 to some further research. We remark that the modification of the parametrix method, developed in [3, 6], provi- des, under more restrictive conditions on the distribution of {ξk}, the best possible converence rate O ( 1√ n ) . 2. Applications. In this section, we formulate two applications of Theorem 1.1. The proofs are given in Section 4. 2.1. Mixing and convergence rates for difference approximations to stochastic di- fferential equations. Under condition (B1) and some recurrence conditions, the process X is ergodic, i.e., possesses a unique invariant distribution µinv (see [8, 9]). Moreover, an explicit estimates for the β-mixing coefficients and for the rate of convergence of Px,t ≡ P(X(t) ∈ ·\|X(0) = x) to µinv in total variation norm are also available. The processes Xn, restricted to 1 n Z+, are a Markov chains. The following natural question takes its origins in a numerical applications: can the mentioned above estimates for the mixing and convergence rate be made uniform over the class {Xn, n ≥ 1, X}? This question is studied in the recent paper [4], see more discussion therein. In this subsection, we use the truncated local limit theorem (Theorem 1.1) in order to establish the required uniform estimates. Denote, by ‖ · ‖var, the total variation norm. Recall that the β-mixing coefficient for X is defined by βx(t) ≡ sup s∈R+ E ‖P(·\|Fs 0, X(0) = x)− P(·\|X(0) = x)‖var,F∞t+s , t ∈ R+, where Fb a ≡ σ(X(s), s ∈ [a, b]), P(·\|Fs 0, X(0) = x) denotes the conditional distribution of the process X with X(0) = x w.r.t. Fs 0, and ‖κ‖var,G df= sup B1∩B2=∅, B1∪B2=C(R+,Rm) B1,B2∈G [ κ(B1)− κ(B2) ] . The β-mixing coefficient {βn x (t), t ∈ 1 n Z+} for Xn is defined analogously. Theorem 2.1. Let conditions (B1) and (B3) hold true. Suppose also that (B4) there exists R0 > 0 and r > 0 such that (a(x), x)Rd ≤ −r‖x‖, ‖x‖ ≥ R0; (B5) there exists κ > 0: E exp[κ‖ξ‖] < +∞. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 346 A. M. KULIK Then, for every process X, Xn, n ≥ 1, there exists unique invariant distribution µinv, µ n inv. Moreover, there exist n0 ∈ N, a function C : Rd → R+ and a constant c > 0 such that ‖Pn x,t − µn inv‖var ≤ C(x)e−ct, t ∈ 1 n Z+, n ≥ n0, ‖Px,t − µinv‖var ≤ C(x)e−ct, t ∈ R+, βn x (t) ≤ C(x)e−ct, t ∈ 1 n Z+, n ≥ n0, βx(t) ≤ C(x)e−ct, t ∈ R+. Remarks. 2.1. The statement of Theorem 2.1 is analogous to the one of Theorem 1 [4]. The main improvement is that the conditions (D1) – (D3) of Theorem 1 [4] are replaced by (seemingly, the mildest possible) condition (B3). In addition, Theorem 2.1, unlike Theorem 1 [4], admits non-constant diffusion coefficients b. 2.2. The mixing and convergence rates established in Theorem 2.1 are called an exponential ones. If the recurrence condition (B4) is replaced by a weaker ones, then the subexponential or polynomial rates can be established (see Theorem 1 [4], cases 2 and 3). We do not give an explicit formulation here in order to shorten the exposition. 2.2. Difference approximation for local times of multidimensional diffusions. Consider a W-measure µ on Rd, that is, by definition [10] (Chapter 8), a σ-finite measure satisfying the condition sup x∈Rm ∫ ‖y−x‖≤1 wd(‖y − x‖)µ(dy) < +∞ with wd(r) =  r, d = 1, max(− ln r, 1), d = 2, r2−d, d > 2. (2.1) Every such a measure generates a W-functional [10] (Chapter 6) of a Wiener process W on Rd, ϕs,t = ϕs,t(W ) = t∫ s dµ dλd (W (r)) dr, 0 ≤ s ≤ t. (2.2) For singular µ, equality (2.2) is a formal notation, that can be substantiated via an approximative procedure with µ approximated by an absolutely continuous measures [10] (Chapter 8). The functional ϕ is naturally interpreted as the local time for the Wiener process, correspondent to the measure µ. Next, let the process X be defined by (0.1) and satisfy (0.4), that means that the asymptotic behavior of its transition probability density as t→ 0+ is similar to the one of the transition probability density for the Wiener process. Then the estimates, analogous to those given in [10] (Chapter 8) provide that the W -functional of the process X ϕs,t = ϕs,t(X) = t∫ s dµ dλd (X(r)) dr, 0 ≤ s ≤ t, (2.3) is well defined. We interpret this functional as the local time for the diffusion process X, correspondent to the measure µ. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 347 At last, let the sequence Xn, n ∈ N of difference approximations for X be defined by (0.2), (0.3). Consider a sequence of the functionals ϕn(Xn) of the processes Xn of the form ϕs,t n = ϕs,t n (Xn) df= 1 n ∑ k : s≤ k n <t Fn ( Xn ( k n )) , 0 ≤ s < t. (2.4) In Theorem 2.2 below, we establish sufficient conditions for the joint distributions of (ϕn, Xn) to converge weakly to the joint distribution of (ϕ,X). Thus, it is natural to say that the functionals ϕn defined by (2.4) provide the difference approximation for the local time ϕ defined by (2.3). For the further discussion and references concerning this problem, we refer the reader to the recent paper [5]. We fix x ∈ Rd and suppose that Xn(0) = X(0) = x. We denote T = { (s, t) : 0 ≤ ≤ s ≤ t } . In order to shorten exposition, we suppose µ to be finite and to have a compact support. Together with the functionals ϕn that are discontinuous w.r.t. variables s, t, we consider the “random broken line” processes ψs,t n = ϕ j−1 n , k−1 n n − (ns− j + 1)ϕ j−1 n , j n n + (nt− k + 1)ϕ k−1 n , k n n , s ∈ [ j − 1 n , j n ) , t ∈ [ k − 1 n , k n ) . Theorem 2.2. Let conditions (B1), (B6 2), (B3) hold true. Suppose also that (B6) Fn(x) ≥ 0, x ∈ Rd, n ≥ 1 and 1 n sup x∈Rd Fn(x) → 0, n→∞; (B7) measures µn(dx) ≡ Fn(x)λd(dx) weakly converge to µ; (B8) lim δ↓0 lim sup n→+∞ sup x∈Rd ∫ ‖y−x‖≤δ wd(‖y − x‖)µn(dy) → 0. Then (Xn, ψn(Xn)) ⇒ (X,ϕ(X)) in a sense of weak convergence in C(R+,Rd)× × C(T,R+). Remarks. 2.3. The statement of Theorem 2.2 is analogous to the one of Theorem 2.1 [5]. The main improvement is that the condition A3) of Theorem 2.1 [5] is replaced by (seemingly, the mildest possible) condition (B3). 2.4. Once Theorem 2.2 is proved, one can use the standard truncation procedure in order to replace the moment condition (B6 2) by the Lyapunov type condition “∃ δ > > 0: E‖ξk‖2+δ < +∞” (e.g. [11], Section 5). 2.5. For examples and a discussion on the relation between conditions (2.1) and (B8), we refer the reader to [5]. 3. Partial Malliavin calculus on a space with a product measure. For every given n ∈ N and t ∈ [0, 1], the value Xn(t) is a functional of ξ1, . . . , ξn and thus can be interpreted as a functional on the space (Rd)n with the product measure µn. However, under the conditions of Theorem 1.1, µ may contain a singular component and therefore it may fail to have logarithmic derivative. Thus, in general, one can not write the integration-by-parts formula on the probability space ( (Rd)n, (B(Rd))⊗n, µn ) . We overcome this difficulty by using the following trick. Under condition (B3), the measure µ can be decomposed into a sum ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 348 A. M. KULIK µ = απU + (1− α)ν, (3.1) where πU is the uniform distribution on U. One can write (on an appropriate probability space) the representation for {ξk} corresponding to (3.1): ξk = εkηk + (1− εk)ζk, (3.2) where ηk ∼ πU , ζk ∼ ν, and the distribution κ of εk is equal to Bernoulli distribution with κ{1} = α. This representation allows one to consider the family ξ1, . . . , ξn (and, therefore, the process Xn) as a functional on the following probability space: Ω = (Rd×{0, 1}×Rd)n, F = (B(Rd)⊗2{0,1}⊗B(Rd))n, P = (πU×κ×ν)n. (3.3) Now, the measure πU has a logarithmic derivative w.r.t. a properly chosen vector field, and some kind of an integration-by-parts formula can be written on the probability space (Ω,F,P) (see Subsection 3.1 below). The Malliavin-type calculus, associated to this formula, is our main tool in the proof of Theorems 1.1, 1.2. We call this calculus a partial one because the stochastic derivative, this calculus is based on, is defined w.r.t. a proper group of variables, while the other variables play the role of interfering terms. In this section, we give the main constructions of the partial Malliavin calculus, associated to the representation (3.2). 3.1. Integration-by-parts formula. Derivative and divergence. Sobolev classes. Denote Ω = Ω1 × Ω2 × Ω3, Ω1 = Ω3 = (Rd)n, Ω2 = {0, 1}n. We write a point ω ∈ Ω in the form ω = (η, ε, ζ), where η = (η1, . . . , ηn) ∈ (Rd)n, ε = (ε1, . . . , εn) ∈ {0, 1}n, ζ = (ζ1, . . . , ζn) ∈ (Rd)n and ηk = (ηk1, . . . , ηkd), ζk = (ζk1, . . . , ζkd). In this notation, the random variables ηk, εk, ζk are defined just as the coordinate functionals: ηk(ω) = ηk, εk(ω) = εk, ζk(ω) = ζk, ω = (η, ε, ζ) ∈ Ω. Denote by C the set of bounded measurable functions f on Ω such that, for every (ε, ζ) ∈ Ω2 × Ω3, the function f(·, ε, ζ) belongs to the class C∞(Rd) and ess sup η,ε,ζ ∥∥∥[∇η]jf(η, ε, ζ) ∥∥∥ < +∞, j ∈ N, where ∇η denotes the gradient w.r.t. variable η. For f ∈ C and k = 1, . . . , n, r = 1, . . . , d, denote by ∂krf the derivative of f w.r.t. the variable ηkr. Also, denote by H the space Rd×n considered as a (finite-dimensional) Hilbert space with the usual Euclid norm, and by {ekr, r = 1, . . . , d, k = 1, . . . , n} the canonical basis in it: all coordinates of the vector ekr are equal to zero except the coordinate with the index kr being equal to one. For a given functions ψ : Rd → R and θn : (Rd)n → [0, 1], define the stochastic gradient D by the formula ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 349 [Df ](η, ε, ζ) = θn(ζ) ∑ k,r ψ(ηk)[∂krf ](η, ε, ζ)ekr, f ∈ C. (3.4) This definition can be naturally extended to the functionals taking their values in a finite-dimensional Hilbert space Y (actually, in any separable Hilbert space, but we do not need such a generality in our further construction). Given an orthonormal basis {yl} in Y, denote by CY the set of the functions of the type y = ∑ l flyl, {fl} ⊂ C, and put for such a function Dy = ∑ l [Dfl]⊗ yl. It is easy to see that the definitions of the class CY and the derivative D do not depend on the choice of the basis {yl}. By the construction, D satisfies the chain rule: for any two spaces Y, Z and for any f1, . . . , fm ∈ CY , F ∈ C∞(Y m, Z), m ≥ 1, F (f1, . . . , fm) ∈ CZ and D [ F (f1, . . . , fm) ] = m∑ j=1 [∂jF ](f1, . . . , fm)Dfj . (3.5) We denote D0f = f,D1f = Df. The higher derivatives Dj , j > 1, are defined iteratively: Dj = D · . . . ·D︸︷︷︸ j (note that the first operator in this product acts on the elements of CY while the last one acts on the elements of CH⊗(j−1)⊗Y ). Everywhere below, we suppose that U is an open ball B(z, r) (this obviously does not restrict generality). We define the function ψ in (3.4) by ψ(x) = r2 − ‖x − z‖2. Due to this choice, ψ ∈ C∞(Rd) and ψ = 0 on ∂U. These properties of ψ imply the following integration by parts formula:∫ U [∂rf ](x)ψ(x) dx = − ∫ U [∂rψ](x)f(x) dx, f ∈ C1(Rd), r = 1, . . . , d. As a corollary of this formula, we obtain the following statement. Proposition 3.1. For every h ∈ H and every f ∈ C, the following integration-by- parts formula holds true: E(Df, h)H = −E(ρ, h)Hf, ρ ≡ θn(ζ) ∑ k,r [∂rψ](ηk)ekr. (3.6) The formula (3.6) allows one to introduce, in a standard way, the divergence operator corresponding to the derivative D. For g ∈ CH⊗Y , put δ(g) = − ∑ k,r,l [ (ρ, ekr)gkrl + (Dgkrl, ekr)H ] yl, gkrl = (g, ekr ⊗ yl)H⊗Y . (3.7) By the choice of the function ψ, δ(g) ∈ CY as soon as g ∈ CH⊗Y . The chain rule (3.5) and the integration-by-parts formula (3.6) imply that the operators D and δ are mutually adjoint in a sense of the following duality formula: ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 350 A. M. KULIK E(Df, g)H = Efδ(g), f ∈ CY , g ∈ CY,H . (3.8) Since, for every p ≥ 1 and every Y, CY is dense in Lp(Ω,P, Y ), the duality formula (3.8) provides that, for any p ≥ 1 and any Y, the operatorsD, δ are closable as densely defined unbounded operators D : Lp(Ω,P, Y ) → Lp(Ω,P,H ⊗ Y ), δ : Lp(Ω,P,H ⊗ Y ) → Lp(Ω,P, Y ). Definition 3.1. The Sobolev class Wm p (Y ), p ≥ 1, m ∈ Z+, is the completion of the class CY w.r.t. the norm ‖f‖p,m ≡  m∑ j=0 E‖Djf‖p H⊗j⊗Y 1 p < +∞. Since D is closable in Lp sense, there exists the canonical embedding of Wm p (Y ) into Lp(Ω,P, Y ). We denote W∞ ∞ (Y ) = ⋂ m,pW m p (Y ). If Y = R then we denote the corresponding Sobolev spaces simply by Wm p . 3.2. Algebraic relations for derivative and divergence. Moment estimates. Let us introduce some notation. We denote by C a constant such that its value can be calculated explicitly, but this calculation is omitted. The value of C may vary from line to line. If the value of the constant C depends on some parameters, say m, d, then we write C(m, d). The latter notation indicates that the value of the constant does not depend on other parameters (for instance, n). If, in a sequel, the constant C is referred to, then we endow it with the lower index like C1,C2, etc. We use standard notation {δjk, j, k ∈ N} for the Kronecker’s symbol. For an H ⊗H ⊗ Y -valued element K, we denote by K∗ the element such that (K∗, h⊗ g ⊗ y)H⊗H⊗Y = (K, g ⊗ h⊗ y)H⊗H⊗Y , h, g ∈ H, y ∈ Y. For an X⊗Y -valued element g1 and X⊗Z-valued element g2, we denote by (g1, g2)X the Y ⊗ Z-valued element (g1, g2)X ≡ ∑ l1,l2,l3 (g1, xl1 ⊗ yl2)X⊗Y (g2, xl1 ⊗ zl3)X⊗Z [yl2 ⊗ zl3 ], here {xl}, {yl}, {zl} are orthonormal bases in X, Y and Z, correspondingly. We also denote for an Y ⊗X-valued element g1 and Z ⊗X-valued element g2 (g1, g2)X ≡ ∑ l1,l2,l3 (g1, yl2 ⊗ xl1)Y⊗X(g2, zl3 ⊗ xl1)Z⊗X [yl2 ⊗ zl3 ]. Although the same notation (·, ·)X is used for two slightly different objects, it does not cause misunderstanding further. Consider the L(H)-valued random element (i.e., random operator in H) B, defined by the relations ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 351( Bek1r1 , ek2r2 ) H = − ( D(ρ, ek1r1)H , ek2r2 ) H = −θ2n(ζ)δk1k2 [∂r1∂r2ψ](ηk1)ψ(ηk1), (3.9) k1,2 = 1, . . . , n, r1,2 = 1, . . . , d. We define the action of B on H ⊗ Y -valued element g by Bg = ∑ k,r,l (g, ek,r ⊗ yl)H⊗Y [ [Bek,r]⊗ yl ] . Using representation (3.7), one can deduce the commutation relations for the operators D, δ, analogous to those for the stochastic derivative and integral for the Wiener process (the proof is straightforward and omitted; for the Wiener case, see [12], § 1.2). Proposition 3.2. I. If f ∈ C, g ∈ CH⊗Y , then f · g ∈ CH⊗Y and δ(f · g) = f · δ(g)− (Df, g)H . II. If g ∈ CH⊗Y , then D [ δ(g) ] = Bg + δ ( [Dg]∗ ) . III. If g1, g2 ∈ CH , then (D[δ(g1)], g2)H = (Bg1, g2)H + δ ( (Dg1, g2)H ) + ([Dg1]∗, Dg2)H⊗H . The main result of this subsection is given by the following lemma. Lemma 3.1. Let m, l ∈ N, g ∈W 2m+l−1 2m (H). Then there exists δ(g) ∈W l 2m and ‖δ(g)‖2m,l ≤ C(m, l, d, ψ)‖g‖2m,2m+l−1. (3.10) Remarks. 3.1. On the Wiener space, the typical way to prove estimates of the type (3.10) is to use Meyer’s inequalities for the generator L = δD of the Ornstein – Uhlenbeck semigroup (see, for instance, [12], § 2.4). Moreover, on the Wiener space, (3.10) can be made more precise: the similar inequality holds with 2m+ l− 1 replaced by l+1. In our settings, it is not clear whether the operator δ ·D provides the analogues of Meyer’s inequalities, since it does not have the specific structural properties of the Ornstein – Uhlenbeck generator (such as Mehler’s formula, hypercontractivity of the associated semigroup, etc.). Thus we prove (3.10) straightforwardly by using an iterative integration-by-parts procedure. 3.2. Throughout the exposition, the function ψ is fixed together with the set U = = B(x, z). However, when the constant C depends on the values of ψ or its derivatives, we indicate it explicitly in the notation for C. In order to prove Lemma 3.1, we need some auxiliary statements and notation. For g ∈ CY and m ∈ Z+, we define the random variable \|g\|m by \|g\|m ≡  m∑ j=0 ‖Djg‖2H⊗j⊗Y 1 2 . Lemma 3.2. If g ∈ CH⊗Y then Bg ∈ CH⊗Y and, for every m ∈ Z+, \|Bg\|m ≤ C(m, d, ψ)\|g\|m. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 352 A. M. KULIK Proof. Write Bg in the coordinate form: Bg = ∑ k,r1,r2,l (g, ekr1 ⊗ yl)H⊗Y bk,r1,r2 [ekr2 ⊗ yl], where bk,r1,r2 = (Bekr1 , ekr2)H (recall that (Bek1r1 , ek2r2)H = 0 as soon as k1 6= k2). Write the Leibnitz formula for the higher derivatives: Dm(Bg) = = ∑ k,r1,r2,l ∑ Θ∈2{1,...,m} ∑ k1,r1,...,km,rm ( D#Θ(g, ekr1 ⊗ yl)H⊗Y , ⊗ i∈Θ ekiri ) H⊗#Θ × × Dm−#Θbkr1r2 , ⊗ i6∈Θ ekiri  H⊗(m−#Θ) [ m⊗ i=1 ekiri ⊗ ekr2 ⊗ yl ] , (3.11) where #Θm denotes the number of elements in the set Θ. We write SΘ = ∑ k,r1,r2,l,k1,r1,...,km,rm ( D#Θ(g, ekr1 ⊗ yl)H⊗Y , ⊗ i∈Θ ekiri ) H⊗#Θ × × Dm−#Θbkr1r2 , ⊗ i6∈Θ ekiri  H⊗(m−#Θ) [ m⊗ i=1 ekiri ⊗ ekr2 ⊗ yl ] and estimate ‖SΘ‖H⊗(m+1)⊗Y . The function ψ belongs to C∞ and is bounded together with all its derivatives on U. Thus, one can deduce from the representation (3.9) and formula (3.4) that ∥∥DMbkr1r2 ∥∥ H⊗M ≤ C(M,d, ψ), M ∈ N. In addition, due to (3.9),Dm−#Θbkr1r2 , ⊗ i6∈Θ ekiri  H⊗(m−#Θ) = 0 as soon as ki 6= k for some i 6∈ Θ. Using these facts, we deduce that ∑ ki∈{1,...,n},ri∈{1,...,d},i6∈Θ Dm−#Θbkr1r2 , ⊗ i6∈Θ ekiri 2 H⊗(m−#Θ) ≤ C(m, d, ψ). Thus ‖SΘ‖2H⊗(m+1)⊗Y ≤ C(m, d, ψ)× × ∑ k,r1,r2,l,ki∈{1,...,n},ri∈{1,...,d},i∈Θ ( D#Θ(g, ekr1 ⊗ yl)H⊗Y , ⊗ i∈Θ ekiri )2 H⊗#Θ ≤ ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 353 ≤ C(m, d, ψ)\|g\|2#Θ ≤ C(m, d, ψ)\|g\|2m. Taking the sum over Θ ∈ 2{1,...,m} and using the Cauchy inequality, we obtain the required statement. The lemma is proved. Proposition 3.3. I. Let g1 ∈ CX⊗Y , g2 ∈ CX⊗Z , then (g1, g2)X ∈ CY⊗Z and∣∣(g1, g2)X ∣∣ m ≤ C(m)\|g1\|m\|g2\|m, m ≥ 0. II. Let g ∈ CY and A ∈ L(Y, Z), then Ag ∈ CZ and \|Ag\|m ≤ ‖A‖\|g\|m, m ≥ 0. The second statement is a straightforward corollary of the chain rule (3.5). The first one can be proved using the Leibnitz formula; the proof is totally analogous to the one of Lemma 3.2, and thus we omit the detailed exposition. Remark 3.3. Taking X = R, we obtain that, for g1 ∈ CY , g2 ∈ CZ , g1⊗g2 ∈ CY⊗Z with \|g1 ⊗ g2\|m ≤ C(m)\|g1\|m\|g2\|m, m ≥ 0. Using iteratively statement II of Proposition 3.2, we obtain that, for g ∈ CH and m ≥ 1, the derivative Dm[δ(g)] can be expressed in the form Dm [ δ(g) ] = Fm(g) + δ ( Gm(g) ) , where Fm(g) ∈ CH⊗m , Gm(g) ∈ CH⊗(m+1) are defined via the iterative procedure F0(g) = 0, G0(g) = g, Gi+1(g) = [ DGi(g) ]∗ , Fi+1(g) = DFi(g) +BGi(g), i ≥ 0. The mapping K 7→ K∗ is an isometry in H⊗H⊗Y, thus statement II of Proposition 3.3 provides that ∣∣Gm(g) ∣∣ j ≤ \|g\|m+j , m, j ≥ 0. (3.12) Using Lemma 3.2, we deduce that∣∣Fm(g) ∣∣ j ≤ C(m, j, d, ψ)\|g\|m+j , m, j ≥ 0. (3.13) Thus, in order to prove inequality (3.10) for g ∈ CH , it is sufficient to prove that E ∥∥δ(g)∥∥2m Y ≤ C(m, d, ψ)E\|g\|2m 2m−1 (3.14) for anym ≥ 1, g ∈ CH⊗Y and arbitrary Hilbert space Y. In order to prove estimate (3.14) we embed it into a larger family of estimates. Consider the following objects. 1. Numbers k0, . . . , kv ∈ Z+, v ≤ 2m, such that k0 + . . . + kv = 2m. Denote Ij = [k0 + . . .+ kj−1 + 1, k0 + . . .+ kj ] ∩N, j = 1, . . . , v, I0 = [1, k0] ∩N (if k0 = 0 then I0 = ∅). 2. Function σ : {1, . . . , 2m} → {1, . . . ,m} such that #σ−1({i}) = 2 and # [ Ij ∩ ∩ σ−1({i}) ] ≤ 1 for every i = 1, . . . ,m and j = 0, . . . , v. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 354 A. M. KULIK Lemma 3.3. Let f0 ∈ CY ⊗k0 , gj ∈ CH⊗Y ⊗kj . Then E ∑ l1,...,lm ( g0, ⊗ i∈I0 ylσ(i) ) Y ⊗k0 v∏ j=1 δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj  ≤ ≤ C(v, d, ψ)E \|g0\|v v∏ j=1 \|gj \|v−1 . (3.15) Remark 3.4. The left-hand side of (3.14) can be rewritten to the form E ∑ l1,...,lm δ ( (g, yl1)Y ) δ ( (g, yl1)Y ) δ((g, yl2)Y )δ ( (g, yl2)Y ) . . . . . . δ ( (g, ylm)Y ) δ ( (g, ylm)Y ) . (3.16) If v = 2m, k0 = 0, k1 = . . . = k2m = 1, g0 = 1, g1 = . . . = g2m = g ∈ CH⊗Y , then the left-hand side of (3.15) coincides with the expression written in (3.16). Thus Lemma 3.3 implies estimate (3.14). Proof of the lemma. We use induction by v. For v = 0, conditions imposed on σ can be satisfied if m = 0, only (i.e., if g0 is a function valued in R). Thus, for v = 0, (3.15) is trivial since g0 ≤ \|g0\| ≡ \|g0\|0. For v = 1, conditions imposed on σ imply that I0 = {1, . . . ,m}, I1 = {m + 1, . . . , 2m} and the function σ, restricted to either I0 or I1, is bijective. Thus the left-hand side of (3.15) can be rewritten to the form E ∑ l1,...,lm ( g0, m⊗ i=1 yli ) Y ⊗m δ (( gj , m⊗ i=1 ylπ(i) ) Y ⊗m ) , where π is some permutation of {1, . . . ,m}. Using duality formula (3.8), we rewrite this as E ( Dg0, Aπg ) H⊗Y ⊗m , where the operator Aπ ∈ L(H ⊗ Y ⊗m) is defined by A[h⊗ yl1 ⊗ . . .⊗ ylm ] = h⊗ ylπ(1) ⊗ . . .⊗ ylπ(m) . (3.17) One can easily see that Aπ is an isometry operator, and thus Proposition 3.3 provides that (3.15) holds true for v = 1 with C(1, d, ψ) = 1. Suppose that, for some V ≥ 2, (3.15) holds true for all v ≤ V − 1. Let us prove that (3.15) holds for v = V, also. For every l1, . . . , lm, take g = ( g1, ⊗ i∈I1 ylσ(i) ) Y ⊗k1 , f = ( g0, ⊗ i∈I0 ylσ(i) ) Y ⊗k0 V∏ j=2 δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj  and apply duality formula (3.8). Then the left-hand side of (3.15) transforms to the form ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 355 E ∑ l1,...,lm D(g0,⊗ i∈I0 ylσ(i) ) Y ⊗k0 , ( g1, ⊗ i∈I1 ylσ(i) ) Y ⊗k1  H × × V∏ j=2 δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj + + V∑ r=2 E ∑ l1,...,lm ( g0, ⊗ i∈I0 ylσ(i) ) Y ⊗k0 × × D δ (gr, ⊗ i∈Ir ylσ(i) ) Y ⊗kr , (g1,⊗ i∈I1 ylσ(i) ) Y ⊗k1  H × × ∏ j∈{2,...,V }\{r} δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj . (3.18) Let us estimate every summand in (3.18) separately. The idea is that every such summand can be written as E ∑ l1,...,lm̃ g̃0,⊗ i∈Ĩ0 ylσ̃(i)  Y ⊗k̃0 v∏ j=1 δ g̃j , ⊗ i∈Ĩj ylσ̃(i)  Y ⊗k̃j  (3.19) with v = V − 1 or v = V − 2 and some new m̃, k̃0, . . . , km̃v, g̃0, g̃v, σ̃, and thus the inductive supposition can be applied. Consider the first summand. Denote by J the set of indices r ∈ {1, . . . ,m} such that σ−1 ( {r} ) ⊂ I0 ∪ I1. In order to shorten notation, we suppose that J = {1, . . . ,#J} (this does not restrict generality since one can make an appropriate permutation of the set {1, . . . ,m} in order to provide such a property). Take permutations π0 : I0 → I0 and π1 : I1 → I1 such that [σ ◦ π0](i) = i, i ∈ {1, . . . ,#J}, [σ ◦ π1](i) = i− k0, i ∈ {k0 + 1, . . . , k0 + #J}. Then the first summand in (3.18) can be rewritten to the form E ∑ l#J+1,...,lm  ∑ l1,...,l#J ( D ( Aπ0g0, k0⊗ i=1 ylσ(π0(i)) ) Y ⊗k0 , ( Aπ1g1, k0+k1⊗ i=k0+1 ylσ(π1(i) ) Y ⊗k1  H  V∏ j=2 δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj , where the operators Aπ0 and Aπ1 are defined analogously to (3.17). Denote ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 356 A. M. KULIK g̃0 = (DAπ0g0, Aπ1g1)H⊗Y ⊗(#J) , then ∑ l1,...,l#J D(Aπ0g0, k0⊗ i=1 ylσ(π0(i)) ) Y ⊗k0 , ( Aπ1g1, k0+k1⊗ i=k0+1 ylσ(π1(i) ) Y ⊗k1  H  = = g̃0,  k0⊗ i=#J+1 ylσ(π0(i)) ⊗  k1⊗ i=k0+#J+1 ylσ(π1(i))  Y k0+k1−2#J . Put m̃ = m −#J, k̃0 = k0 + k1 − 2#J, k̃1 = k2, . . . , k̃V−1 = kV and let {Ĩ0, Ĩ1, . . . . . . , ĨV−1} be the partition of {1, . . . , 2m̃} corresponding to the family {k̃0, . . . , k̃V−1}. Put g̃1 = g2, . . . g̃V−1 = gV (g̃0 is already defined). At last, define function σ̃ by σ̃(i) =  σ(π0(i+ #J)), i = 1, . . . , k0 −#J, σ(π1(i+ 2#J)), i = k0 −#J + 1, . . . , k0 + k1 − 2#J, σ(i+ 2#J), i = k0 + k1, . . . , 2m̃. Under such a notation, the first summand in (3.18) has exactly the form (3.19) with v = V − 1, and the inductive supposition provides that this summand is dominated by the term C(V − 1, d, ψ)E \|g̃0\|V−1 V−1∏ j=1 \|g̃j \|V−2 . Since Aπ0 , Aπ1 are isometric operators, we can apply Proposition 3.3 and obtain that \|g̃0\|V−1 = ∣∣∣(DAπ0g0, Aπ1g1)H⊗Y ⊗(#J) ∣∣∣ V−1 ≤ ≤ C(V − 1)\|DAπ0g0\|V−1\|Aπ1g1\|V−1 ≤ C(V − 1)\|g0\|V \|g1\|V−1. For every j = 1, . . . , V − 1, \|g̃j \|V−2 = \|gj+1\|V−2 ≤ \|gj+1\|V−1. Thus, under the inductive supposition, the first summand in (3.18) is dominated by the expression given in the right-hand side of (3.15). All the V − 1 summands in the second sum in (3.18) have the same form and can be estimated similarly; let us make such an estimation for r = 2. Using Proposition 3.2, we rewrite this summand to the form E ∑ l1,...,lm ( g0, ⊗ i∈I0 ylσ(i) ) Y ⊗k0 × × B (g2,⊗ i∈Ir ylσ(i) ) Y ⊗kr  , ( g1, ⊗ i∈I1 ylσ(i) ) Y ⊗k1  H × ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 357 × V∏ j=3 δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj + + E ∑ l1,...,lm ( g0, ⊗ i∈I0 ylσ(i) ) Y⊗k0 δ D (g2,⊗ i∈Ir ylσ(i) ) Y ⊗kr , ( g1, ⊗ i∈I1 ylσ(i) ) Y ⊗k1  H  V∏ j=3 δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj + + E ∑ l1,...,lm ( g0, ⊗ i∈I0 ylσ(i) ) Y ⊗k0 D (g2,⊗ i∈Ir ylσ(i) ) Y ⊗kr ∗ , ( Dg1, ⊗ i∈I1 ylσ(i) ) Y ⊗k1  H⊗H V∏ j=3 δ gj , ⊗ i∈Ij ylσ(i)  Y ⊗kj . (3.20) Let us show that, after an appropriate rearrangement of the indices i, every summand in (3.20) can be rewritten to the form (3.19). Such a rearrangement can be organized in the way, totally analogous to the one used before while the first summand in (3.18) was estimated. Therefore, in order to shorten exposition, we do not write here an explicit form for the permutations of the indices, used in such a rearrangement. The first summand in (3.20) has the form (3.19) with v = V − 2, g̃j = gj+2, j = 1, . . . , V − 2, g̃0 = ( Aπ1 [Bg2], Aπ2 [g1 ⊗ g0] ) H⊗Y ⊗#J1 , where J1 is the set of such i ∈ {1, . . . ,m} that σ−1({i}) ⊂ I0∪I1∪I2 (we do not write here an explicit expressions neither for the permutations π1, π2 nor for k̃0, . . . , k̃V−2, σ̃). Under inductive supposition, this summand is estimated by C(V − 2, d, ψ) E\|g̃0\|V−2 V∏ j=3 \|gj \|V−3 ≤ ≤ C(V, d, ψ) E\|Bg2\|V−2\|g1\|V−2\|g0\|V−2 V∏ j=3 \|gj \|V−3 ≤ ≤ C(V, d, ψ)E\|g2\|V−2\|g1\|V−2\|g0\|V−2 V∏ j=3 \|gj \|V−3 ≤ C(V, d, ψ)E\|g0\|V V∏ j=1 \|gj \|V−1, here we used Proposition 3.3 and Lemma 3.2. The second summand in (3.20) has the form (3.19) with v = V − 1, g̃j = gj+1, j = 2, . . . , V − 1, g̃0 = g0, ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 358 A. M. KULIK g̃1 = ( D[Aπ1g2], Aπ2g1 ) H⊗Y ⊗#J2 , where J2 is the set of such i ∈ {1, . . . ,m} that σ−1({i}) ⊂ I1 ∪ I2. This summand again is estimated by C(V − 1, d, ψ) E\|g0\|V−1\|g̃1\|V−2 V∏ j=3 \|gj \|V−2 ≤ ≤ C(V, d, ψ) E\|g0\|V−1\|\|Dg2\|V−2\|g1\|V−2\|g0\|V−2 V∏ j=3 \|gj \|V−3 ≤ ≤ C(V, d, ψ) E\|g0\|V V∏ j=1 \|gj \|V−1. At last, the third summand in (3.20) has the form (3.19) with v = V − 2, g̃j = gj+2, j = 1, . . . , V − 2, g̃0 = ( Aπ1g0, ([DAπ2g2] ∗, DAπ3g2)H⊗H ) Y ⊗#J1 , and again is estimated by C(V − 2, d, ψ) E\|g̃0\|V−2 V∏ j=3 \|gj \|V−3 ≤ ≤ C(V, d, ψ) E\|g0\|V−2\|Dg2\|V−2\|Dg1\|V−2 V∏ j=3 \|gj \|V−3 ≤ ≤ C(V, d, ψ) E\|g0\|V V∏ j=1 \|gj \|V−1. The estimates given above show that (3.15) holds for v = V as soon as it holds for v = V − 2 and v = V − 1. We have already proved that (3.15) holds for v = 0, 1. Thus, (3.15) holds for every v. The lemma is proved. Proof of Lemma 3.1. We have already proved (3.10) to hold for every g ∈ CH . Now, let g ∈ W 2m+l−1 2m (H). Consider {gn} ⊂ CH such that gn → g in W 2m+l−1 2m (H) (recall that CH is dense in any W k p (H) by definition). By (3.10), for any k = 0, . . . , l, ∥∥Dkδ(gn)−Dkδ(gN ) ∥∥ L2m(Ω,P,H⊗k) ≤ ≤ C(m, l, d, ψ)‖gn − gN‖2m,2m+l−1 → 0, n, N → +∞. Thus there exist Fk ∈ L2m(Ω,P,H⊗k), k = 0, . . . , l, such that ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 359∥∥Dkδ(gn)− Fk ∥∥ L2m(Ω,P,H⊗k) → 0, n→ +∞, k = 0, . . . , l. Since operator δ is closed, F0 = δ(g). Using that operator D is closed, one can veri- fy inductively that Fk = DFk−1, k = 1, . . . , l. This means that δ(g) ∈ W l 2m with Dkδ(g) = Fk, k = 0, . . . , l. At last, using (3.10) we get ‖δ(g)‖2m 2m,l = E l∑ k=0 ‖Fk‖2m H⊗k ≤ lim sup n E l∑ k=0 ∥∥Dkδ(gn) ∥∥2m H⊗k = = lim sup n ‖δ(gn)‖2m 2m,l ≤ C(m, l, d, ψ) lim sup n ‖gn‖2m 2m,3m−1 = = C(m, d, ψ)‖g‖2m 2m,2m+l−1. The lemma is proved. 3.3. Malliavin’s representation for the densities of the truncated distributions of smooth functionals. The typical result in the Malliavin calculus on the Wiener space is that, when the components f1, . . . , fd of a random vector f = (f1, . . . , fd) are smooth enough and the Malliavin matrix σf = { (Dfi, Dfj)H }d i,j=1 is non-degenerate in a sense that [detσf ]−1 ∈ ⋂ p≥1 Lp(Ω,F,P), (3.21) the distribution of f has a smooth density (see, for instance, [12], § 3.2). Such kind of a result is useless in the framework, introduced in Subsection 3.1, since there does not exist any functional f satisfying (3.21): if ε1 = . . . = εn = 0 then Df = 0 for every f ∈ C. In order to overcome this difficulty we use the following truncation procedure: we consider, instead of P, a new (non-probability) measure PΞ(·) = P(· ∩Ξ) with some set Ξ ∈ σ(ε, ζ). If this set is chosen in such a way that (3.21) holds true with P replaced by PΞ then the Malliavin’s calculus can be applied in order to investigate the law of f w.r.t. PΞ. In this subsection, we give the Malliavin’s representation for the density of this law. All principal steps in our consideration are analogous to those in the standard Malliavin calculus on the Wiener space (see, for instance, [12], Chapter 3). Therefore, we sketch the proofs only. Let f1, . . . , fd ∈ C be fixed, consider the Malliavin matrix σf = (σf ij) d i,j=1, σf ij = (Dfi, Dfj)H = ∑ k,r ψ(ηk) [ ∂krfi(η, ε, ζ) ][ ∂krfj(η, ε, ζ) ] . Consider a set Ξ ∈ σ(ε, ζ) such that Ξ ⊂ {detσf > 0} and E1IΞ[detσf ]−p <∞, p ≥ 1. (3.22) Then 1Ξ ∈ C and D1Ξ = 0. Put %f,Ξ(ω) = [σf (ω)]−1, ω ∈ Ξ, 0, ω 6∈ Ξ. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 360 A. M. KULIK Proposition 3.4. %f,Ξ ∈W∞ ∞ (Rd×d) and (D%f,Ξ, h)H = −%f,Ξ(Dσf , h)H% f,Ξ, h ∈ H. (3.23) Sketch of the proof. It is enough to prove that %f,Ξ ∈ ⋂ p≥1W 1 p (Rd×d) and (3.23) holds true. Suppose that σf ≥ cIRd with some c > 0. Then one can easily see that %f,Ξ ∈ C and (3.23) follows from the well known formula for the derivative of the inverse matrix, d dt [ A(t) ]−1 = − [ A(t) ]−1 [ d dt A(t) ] [ A(t) ]−1 . In the general case, consider the matrix-valued functions σf,c = σf + cIRd and %f,Ξ,c = = 1IΞ · [σf,c]−1, c > 0. Condition (3.22) provides that %f,Ξ,c → %f,Ξ, c → 0+ in any Lp. It is already proved that (3.23) holds true for the functionals indexed by c. Thus, passing to the limit as c→ 0+, we obtain the required statement. Denote ϑf,Ξ i = ∑d k=1 %f,Ξ ki Dfk, i = 1, . . . , d. Also denote, by EΞ, the expectation w.r.t. PΞ. Proposition 3.5. For every i = 1, . . . , d, υ ∈W∞ ∞ and every F ∈ C∞b (Rd), EΞ[∂iF ](f1, . . . , fd)υ = EΞF (f1, . . . , fd)δ ( υϑf,Ξ i ) . (3.24) Sketch of the proof. It follows from Propositions 3.3, 3.4 and Lemma 3.1 that υ · ϑf i ∈ Dom(δ). Since Ξ ∈ σ(ε, ζ), the function 1IΞ belongs to C and has its stochastic derivative equal to 0. Proposition 3.2 provides that δ(1IΞg) = 1IΞδ(g), g ∈ Dom(δ). Therefore E[∂iF ](f1, . . . , fd)1IΞυ = d∑ j=1 E[∂jF ](f1, . . . , fd)1IΞυ[σf%f,Ξ]ij = = d∑ j=1 d∑ k=1 E[∂jF ](f1, . . . , fd)1IΞυ% f,Ξ ki σ f jk = = E d∑ k=1 1IΞυ% f,Ξ ki  d∑ j=1 [∂jF ](f1, . . . , fd)Dfj , Dfk  H = = E ( D[F (f1, . . . , fd)], 1IΞυϑ f,Ξ i ) H = = EF (f1, . . . , fd)δ ( 1IΞυϑ f i ) = EF (f1, . . . , fd)1IΞδ ( υϑf i ) , that provides (3.24). Put υf,Ξ 1 = 1, υf,Ξ l+1 = δ ( υf,Ξ l ϑf,Ξ l ) , l = 1, . . . , d, Υf,Ξ = υf,Ξ d+1, Υf,Ξ i = δ ( Υf,Ξϑf,Ξ i ) , i = 1, . . . , d. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 361 Write Pf Ξ for the distribution of f w.r.t. PΞ. For α1, . . . , αd ∈ {0, 1}, denote 1Iα1...αd (x) = 1I(−1)α1x1≥0,...,(−1)αdxd≥0, x ∈ Rd. Proposition 3.6. The distribution Pf Ξ has a density pf Ξ, bounded together with all its derivatives ∂ip f Ξ, i = 1, . . . , d. For any α1, . . . , αd ∈ {0, 1}, pf Ξ(y) = (−1)α1+...+αdE1Iα1...αd (f − y)Υf,Ξ, (3.25) ∂ip f Ξ(y) = (−1)α1+...+αd+αi+1E1Iα1...αd (f − y)Υf,Ξ i , (3.26) y ∈ Rd. Sketch of the proof. Applying iteratively (3.24) one can deduce that, for every F ∈ C∞b (Rd), EΞ[∂1 . . . ∂dF ](f1, . . . , fd) = EF (f1, . . . , fd)Υf,Ξ, (3.27) EΞ[∂1 . . . ∂d∂iF ](f1, . . . , fd) = EF (f1, . . . , fd)Υ f,Ξ i , i = 1, . . . , d. (3.28) Now, the informal way to get representation (3.25) is to apply (3.27) to F = 1Iα1...αd : pf Ξ(y) = (−1)α1+...+αd+d∂1 . . . ∂dEΞ1Iα1...αd (f − y) = = (−1)α1+...+αdEΞ[∂1 . . . ∂d1Iα1...αd ](f − y) = = (−1)α1+...+αdE1Iα1...αd (f − y)Υf,Ξ. (3.29) In order to justify (3.29) one should consider smooth approximations Fn for the function F = 1Iα1...αd and use Fubini theorem (we omit detailed exposition here, referring the reader, for instance, to [12], § 3.1, 3.2). Similarly, (3.26) is provided by (3.28) and the formula ∂ip f Ξ(y) = (−1)α1+...+αd+1EΞ[∂1 . . . ∂d∂i1Iα1...αd ](f − y). 3.4. Estimates for the densities of the truncated distributions of smooth functionals. Proposition 3.6 immediately provides the following family of estimates for the density pf Ξ of the truncated distribution of f. Corollary 3.1. For any y ∈ Rd, pf Ξ(y) ≤ ‖Υf,Ξ‖L2 min α1,...,αd∈{0,1} P 1 2 Ξ ( (−1)α1f1 ≥ (−1)α1y1, . . . . . . , (−1)αdfd ≥ (−1)αdyd ) ≤ ‖Υf,Ξ‖L2 , ∂ip f Ξ(y) ≤ ‖Υf,Ξ i ‖L2 min α1,...,αd∈{0,1} P 1 2 Ξ ( (−1)α1f1 ≥ (−1)α1y1, . . . . . . , (−1)αdfd ≥ (−1)αdyd ) ≤ ‖Υf,Ξ i ‖L2 , i = 1, . . . , d. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 362 A. M. KULIK In particular, pf Ξ satisfies Lipschitz condition with the constant L = ∑d i=1 ‖Υf,Ξ i ‖L2 . In this subsection we give explicit estimates for ‖Υf,Ξ‖L2 , ‖Υ f,Ξ i ‖L2 , i = 1, . . . , d. Our estimates somewhat differ from the standard Malliavin-type ones. In our consi- derations, we operate with the matrix [σf ]−1 straightforwardly and do not use (unlike in the standard Malliavin’s approach) representation of this matrix via the Cramer’s formula [σf ]−1 = [detσf ]−1Σf (Σf denotes the cofactor matrix for σf ). This is caused by our goal to prove, together with existence of the density, an explicit estimates for it like the estimate (ii) of Theorem 1.1. Let us give an iterative description of the family {υf,Ξ l } involved into construction of Υf,Ξ, Υf,Ξ i . We introduce two families of operators acting on W∞ ∞ : Ii : ϕ 7→ δ(ϕDfi), Jijk : ϕ 7→ ϕ(Dσf jk, Dfi)H , i, j, k = 1, . . . , d. We call any operator I1, . . . , Id an operator of the type I , and any operator from the set {Jijk, i, j, k = 1, . . . , d} an operator of the type J . We denote by K(m,M) the class of all functions that can be obtained from ϕ ≡ 1 by applying, in arbitrary order, of m operators of the type I and M operators of the type J. Proposition 3.7. For any l = 1, . . . , d + 1, there exist constant C(d, l) ∈ N such that υf,Ξ l is a sum of at most C(d, l) summands of the type ϕ r∏ k=1 %f,Ξ ikjk , (3.30) where ik, jk = 1, . . . , d are arbitrary and ϕ belongs to some class K(m,M) with m+M = l − 1 and r = M + l − 1. Proof. We use induction by l. For l = 1, the statement is trivial since υf,Ξ 1 = 1 ∈ ∈ K(0, 0). Suppose the statement of the Lemma to hold true for some l ≤ d. Let us prove this statement for l+1. Due to the inductive supposition, υf,Ξ l is a sum of at most C(d, l) summands of the type δ ( ϕ r∏ k=1 %f,Ξ ikjk ϑf,Ξ l ) , ϕ ∈ K(m,M), m+M = l, M + l = r. We have δ ( ϕ r∏ k=1 %f,Ξ ikjk ϑf,Ξ l ) = d∑ q=1 δ ( ϕ [ r∏ k=1 %f,Ξ ikjk ] %f,Ξ ql Dfl ) . Thus, υf,Ξ l is a sum of at most dC(d, l) summands of the type δ ( ϕ r+1∏ k=1 %f,Ξ ikjk Dfl ) , ϕ ∈ K(m,M), m+M = l, M + l = r. Due to Propositions 3.2 and 3.4, ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 363 δ ( ϕ r+1∏ k=1 %f,Ξ ikjk Dfl ) = δ(ϕDfl) r+1∏ k=1 %f,Ξ ikjk − ( D r+1∏ k=1 %f,Ξ ikjk , ϕDfl ) H = = δ(ϕDfl) r+1∏ k=1 %f,Ξ ikjk − l+1∑ q=1 ϕ  ∏ k≤r+1, k 6=q %f,Ξ ikjk (%f,Ξ(Dσf , Dfl)%f,Ξ ) iqjq . (3.31) Since ϕ ∈ K(m,M), δ(ϕDfl) ∈ K(m + 1,M). Thus, the first term in the right-hand side of (3.31) has the form (3.30). Every summand in the sum in the right-hand side of (3.31) is a sum of d2 terms of the type ϕ(Dσf ĩj̃ , Dfl)H [ r+2∏ k=1 %f,Ξ ĩk j̃k ] . Every such a term has the form (3.30), since ϕ(Dσf ĩj̃ , Dfl)H ∈ K(m,M + 1) for ϕ ∈ ∈ K(m,M). Therefore, the statement of the Lemma holds true for l + 1, also, with C(d, l + 1) = C(d, l) [ 1 + d2(l + 1) ] . The proposition is proved. Recall that Υf,Ξ = υf,Ξd+1 , and thus Proposition 3.7 provides that Υf,Ξ is a sum of not more than C(d) summands of the type (3.30) with ϕ ∈ K(d−M,M) and r = M+d (M may vary from 0 to d). For every such a summand,∥∥∥∥∥ϕ M+d∏ k=1 %f,Ξ ikjk ∥∥∥∥∥ L2 ≤ ‖ϕ‖L4 [ E‖%f,Ξ‖4(M+d) M ] 1 4 , (3.32) where ‖A‖M = maxi,j=1,...,d \|Aij \|, A ∈ Rd×d. Thus, in order to estimate ‖Υf,Ξ‖L2 , it is sufficient to estimate maxϕ∈K(d−M,M) ‖ϕ‖L4 . Denote αi = Dfi, βijk = = (Dσf jk, Dfi)H . Proposition 3.8. For every ϕ ∈ K(d−M,M), ‖ϕ‖L4 ≤ C(d, ψ) ( max i ‖αi‖2(d+1)(d+2),(d+1)2−1 )d−M × × ( max ijk ‖βijk‖2(d+1)(d+2),(d+1)2−1 )M . (3.33) Proof. By Lemma 3.1 and Proposition 3.3, for any i = 1, . . . , d, ϕ ∈W∞ ∞ , m ≥ 0∥∥δ(ϕαi) ∥∥ 2m+2,m2−1 ≤ C(m, d, ψ)‖ϕαi‖2m+2,(m+1)2−1 ≤ ≤ C(m, d, ψ)‖ϕ‖2m+4,(m+1)2−1‖αi‖2(m+1)(m+2),(m+1)2−1. (3.34) By Proposition 3.3, for any i, j, k = 1, . . . , d, ϕ ∈W∞ ∞ , m ≥ 0 ‖ϕβijk‖2m+2,m2−1 ≤ C(m)‖ϕ‖2m+4,m2−1βijk‖2(m+1)(m+2),m2−1 ≤ ≤ C(m, d, ψ)‖ϕ‖2m+4,(m+1)2−1‖βijk‖2(m+1)(m+2),(m+1)2−1. (3.35) Recall that ‖ · ‖L4 = ‖ · ‖4,0 and ϕ ∈ K(d−M,M) is obtained from 1 by applying (in some order) of d −M operators of the type I and M operators of the type J. Thus, in ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 364 A. M. KULIK order to obtain (3.33), one should consequently put m = d, d−1, . . . , 1 and apply either inequality (3.34) or inequality (3.35) depending on what type of the operator (I or J) was applied at this position in the construction of the function ϕ. The proposition is proved. Recall that σf jk = (Dfj , Dfk)H . By Proposition 3.3, \|βijk\|m ≤ C(m)\|Dfi\|m\|Dσf jk\|m ≤ C̃(m)\|Dfi\|m\|Dfj \|m+1\|Dfk\|m+1. Thus, Proposition 3.8 provides the following estimate for ‖Υf,Ξ‖L2 . Denote Nd(f) = max i=1,...,d max m=1,...,(d+1)2 [ E‖Dmfi‖2(d+1)(d+2) H⊗m ] 1 2(d+1)(d+2) . Corollary 3.2. There exists a constant Ld, dependent on d and ψ only, such that ‖Υf,Ξ‖L2 ≤ Ld d∑ M=0 [ Nd(f) ]d+2M[ E‖%f,Ξ‖4(M+d) M ] 1 4 . (3.36) For ‖Υf,Ξ i ‖L2 , the following estimate holds true (we omit the proof since it is totally analogous to the proof of (3.36) given above). Proposition 3.9. For any i = 1, . . . , d, ‖Υf,Ξ i ‖L2 ≤ Ld+1 d+1∑ M=0 [ Nd+1(f) ]d+2M+1[ E‖%f,Ξ‖4(M+d+1) M ] 1 4 . (3.37) At the end of this section, we formulate a general local limit theorem. This theorem is a straightforward corollary of the representation given by Proposition 3.6 and the esti- mates (3.36) and (3.37). Consider the sequence of probability spaces {(Ωn,Fn,Pn), n ≥ ≥ 1} of the type (3.3) with the given measures πU ,κ, ν. Let the function ψ and the functi- ons θn be fixed. Denote Hn = Rd×n and consider the derivative, gradient and Sobolev spaces constructed in Subsection 3.1. For a sequence of random vectors fn : Ωn → Rd and a sequence of sets {Ξn ∈ σ(ε, ζ)} denote by Pn,fn the distribution of fn w.r.t. Pn and by Pn,fn Ξn the distribution of fn w.r.t. PΞn ≡ Pn(· ∩ Ξn). Denote Kd(f,Ξ) = Ld d∑ M=0 [ Nd(f) ]d+2M[ E‖%f,Ξ‖4(M+d) M ] 1 4 . Theorem 3.1. Suppose that {fn} and {Ξn} satisfy condition (C1) fn ∈W (d+2)2 2(d+2)(d+3)(R d) and supn Kd+1(fn,Ξn) < +∞. Then Pn,fn Ξn possess a densities pfn Ξn . Moreover, (a) pfn Ξn (y) ≤ Kd(fn,Ξn)P 1 2 Ξn (‖fn‖ ≥ ‖y‖); (b) pfn Ξn satisfy Lipschitz condition with the common constant equal to d supn Kd+1(fn,Ξn). If, additionally, (C2) fn converge in distribution to some random vector f ; (C3) Pn(Ξn) → 1, n→ +∞; then the distribution of the vector f possess a density pf and ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 365 (c) supy∈Rd \|pfn Ξn (y)− pf (y)\| → 0, n→ +∞. Proof. Although the statements of Corollaries 3.1 and 3.2 are formulated for f ∈ ∈ CRd , they can be extended to f ∈ W (d+2)2 2(d+2)(d+3)(R d) by a standard approximation procedure. For any y ∈ Rd, there exist a choice of the signs α1, . . . , αd such that PΞn ( (−1)α1fn 1 ≥ (−1)α1y1, . . . , (−1)αdfn d ≥ (−1)αdyd ) ≤ PΞn (‖fn‖ ≥ ‖y‖). Therefore, statements (a) and (b) follow immediately from Corollaries 3.1 and 3.2. Statement (b) provides that the sequence {pfn Ξn } has a compact closure in the space C(Rd) with the topology of uniform convergence on a compacts. This together with the conditions (C2), (C3) provides (c). The theorem is proved. 4. Proofs of Theorems 1.1 – 2.2. 4.1. Proof of Theorem 1.1. We reduce the proof of Theorem 1.1 to the verification of the conditions of Theorem 3.1 and explicit estimation of the expression in the right-hand side of (a). We use, without additional discussion, notation introduced in Section 3. Denote fn x,t = Xn(t) − x, where the processes Xn are defined by (0.2), (0.3) with the initial value Xn(0) = x ∈ Rd. When it does not cause misunderstanding, we omit the indices x, t and write fn for fn x,t. We conduct the proof in several steps. First, we give explicit expressions for the derivatives of the functionals fn. Next, we estimate the moments of these derivatives (this allows us to estimateNd+1(fn)). Then, on the properly chosen Ξn, we estimate the inverse matrix for the Malliavin matrix σfn (this allows us to estimate Kd+1(fn,Ξn)). At last, we estimate the tail probabilities PΞn (‖fn‖ ≥ ‖y‖) in order to provide the estimates given in the statement (ii) of Theorem 1.1. Everywhere below we suppose conditions (B1), (Bκ 2 ), (B3) of Theorem 1.1 to hold true. We prove (i) – (iii) in details and give a brief sketch of changes that should be made in order to prove (ii′), (iii′). We put θn(ζ) = 1Imaxk≤n ‖ζk‖≤nς , ς = κ− 1 2κ+ 2 . (4.1) In order to make notation more convenient, we rewrite (0.2) to the form Xn ( k n ) = Xn ( k − 1 n ) + a ( Xn ( k − 1 n )) 1 n + d∑ r=1 br ( Xn ( k − 1 n )) ξkr√ n , (4.2) here ξk1, . . . , ξkd are the components of the vector ξk and b1, . . . , bd are the columns of the matrix b. Lemma 4.1. For every t ∈ [0, 1], Xn(t) ∈ ⋂ p>1W (d+2)2 p (Rd). Derivatives Yn(t) = DXn(t), t ∈ [0, 1], satisfy relations Yn(0) = 0, Yn ( k n ) = Yn ( k − 1 n ) +∇a ( Xn ( k − 1 n )) Yn ( k − 1 n ) 1 n + ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 366 A. M. KULIK + d∑ r=1 [ ∇br ( Xn ( k − 1 n )) Yn ( k − 1 n ) ξkr√ n + + θn(ζ)1Iεk=1ψ(ηk)√ n br ( Xn ( k − 1 n )) ⊗ ekr ] , (4.3) Yn(t) = Yn ( k − 1 n ) + (nt− k + 1) [ Yn ( k n ) − Yn ( k − 1 n )] , (4.4) t ∈ [ k − 1 n , k n ) , k = 1, . . . , n. Sketch of the proof. The proof is quite standard, and thus we just outline its main steps. Using induction by k, one can easily verify that, for every j, k, r, there exists ∂jrXn ( k n ) = ( Yn ( k n ) , ejr ) Hn with Yn defined by (4.3). One can see that Yn ≡ 0 as soon as maxk≤n ‖ζk‖ > nς and, therefore, ess sup ∥∥∥∥Yn ( k n )∥∥∥∥ < +∞ for every k ≤ n. Iterating these considerations, one can verify that ess sup ∥∥∥∥∇m η Xn ( k n )∥∥∥∥ < +∞ for every k ≤ n, m ≤ (d + 2)2, that means that Xn ( k n ) ∈ ⋂ p>1 W (d+2)2 p (Rd) with DXn ( k n ) = Yn ( k n ) , that gives the statement of the Lemma for t = k n . For arbitrary t ∈ [0, 1], this statement holds by linearity. Denote µκ(ξ) = E‖ξ1‖κ. Lemma 4.2. For every p ≥ 1, m ∈ N, there exist constant C(a, b, d, U, µκ(ξ), m, p) such that E‖DmXn(t)‖p H⊗m⊗Rd ≤ C(a, b, d, U, µκ(ξ),m, p)t p 2 , t ∈ [ 1 n , 1 ] . (4.5) Proof. Consider first the case m = 1. It is enough to prove (4.5) for t = k n , k = 1, . . . , n and p = 2q, q ∈ N. We have ∥∥∥∥Yn ( k n )∥∥∥∥2 H⊗Rd = ∥∥∥∥Yn ( k − 1 n )∥∥∥∥2 H⊗Rd + + 1 n ( Yn ( k − 1 n ) , ∇a ( Xn ( k − 1 n )) Yn ( k − 1 n )) H⊗Rd + + d∑ r=1 ( Yn ( k − 1 n ) , ∇br ( Xn ( k − 1 n )) Yn ( k − 1 n )) H⊗Rd ξkr√ n + + 1 n2 ∥∥∥∥∇a(Xn ( k − 1 n )) Yn ( k − 1 n )∥∥∥∥2 H⊗Rd + ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 367 + d∑ r=1 ( ∇a ( Xn ( k − 1 n )) Yn ( k − 1 n ) , ∇br ( Xn ( k − 1 n )) Yn ( k − 1 n )) H⊗Rd ξkr n 3 2 + + d∑ r1,r2=1 ( ∇br1 ( Xn ( k − 1 n )) Yn ( k − 1 n ) , ∇br2 ( Xn ( k − 1 n )) Yn ( k − 1 n )) H⊗Rd ξkr1ξkr2 n + + θn(ζ)1Iεk=1ψ 2(ηk) n d∑ r=1 ∥∥∥∥br (Xn ( k − 1 n ))∥∥∥∥2 Rd , (4.6) here we have used the fact that Yn ( k − 1 n ) , ∇a ( Xn ( k − 1 n )) Yn ( k − 1 n ) , ∇br ( Xn ( k − 1 n )) Yn ( k − 1 n ) , r = 1, . . . , d, belong to the subspace generated by the vectors of the type v ⊗ ejr, v ∈ Rd, j < k, r = 1, . . . , d, and br ( Xn ( k − 1 n )) ⊗ ⊗ ekr, r = 1, . . . , d, are orthogonal to this subspace. Denote ∥∥∥∥Yn ( k n )∥∥∥∥2 H⊗Rd = Υk. Recall that Yn ( k n ) = 0, k = 1, . . . , n, as soon as there exist j = 1, . . . , d such that ‖ζj‖ > nς . Since coefficients a, b are bounded together with their derivatives, we can rewrite (4.6) as Υk = [ Υk−1 + Θk−1 1 n + d∑ r=1 Λk−1,r ξkr√ n 1I‖ζk‖≤nς + + d∑ r1,r2=1 ∆k−1,r1,r2 ξkr1ξkr2 n 1I‖ζk‖≤nς ] 1Imaxj<k ‖ζj‖≤nς , k = 1, . . . , n, with Fk−1 ≡ σ(η1, ε1, ζ1, . . . , ηk−1, εk−1, ζk−1) —- measurable Θk−1, Λk−1,r1,r2 such that \|Θk−1\| ≤ C(a, b, d) [1 + Υk−1] , \|Λk−1,r1,r2 \| ≤ C(a, b, d)Υk−1. (4.7) Since Υk ≥ 0, we have EΥq k ≤ E ( Υk−1 + Θk−1 1 n )q + q−1∑ i=0 q! i!(q − i)! E ( Υk−1 + Θk−1 1 n )i × × [ d∑ r=1 Λk−1,r ξkr√ n 1I‖ζk‖≤nς + d∑ r1,r2=1 ∆k−1,r1,r2 ξkr1ξkr2 n 1I‖ζk‖≤nς ]q−i . (4.8) ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 368 A. M. KULIK We have ξkr = εkηkr + (1 − εk)ζkr and the set U of the possible values of ηk = = (ηk1, . . . , ηkd) is bounded. In addition, \|ζkr\|1I‖ζk‖≤nς ≤ nς . Therefore, E ∣∣∣∣ ξkr√ n ∣∣∣∣l 1I‖ζk‖≤nς ≤ (C(U)nς)l n l 2 = Cl(U)nl(ς− 1 2 ) ≤ Cl(U) n , l ≥ κ+ 1. (4.9) Since E‖ξk‖κ < +∞, E ∣∣∣∣ ξkr√ n ∣∣∣∣l 1I‖ζk‖≤nς ≤ E\|ξkr\|l n l 2 ≤ E(‖ξk‖κ ∨ 1) n , l = 2, . . . , κ. (4.10) Similarly, ∣∣∣∣Eξkr1ξkr2 n 1I‖ζk‖≤nς ∣∣∣∣ ≤ 1 n E‖ξk‖2 ≤ E(‖ξk‖κ ∨ 1) n . (4.11) At last,∣∣∣∣E ξkr√ n 1I‖ζk‖≤nς ∣∣∣∣ = ∣∣∣∣E ξkr√ n 1I‖ζk‖>nς ∣∣∣∣ ≤ n− 1 2 [ E\|ξkr\|κ ] 1 κ [ P(‖ζkr‖ ≥ nς) ]κ−1 κ ≤ ≤ C(U, µκ(ξ))n− 1 2 [ n−ςκ ]κ−1 κ ≤ C(U, µκ(ξ)) n (4.12) (recall that Eξkr = 0). The triple (ηk, εk, ζk) is independent of Fk−1. Thus, taking in (4.8) conditional expectation w.r.t. Fk−1 and taking into account inequalities (4.7), we obtain an estimate EΥq k ≤ E ( Υk−1 + Θk−1 1 n )q + C(a, b, d, U, µκ(ξ), q) 1 + EΥq k−1 n ≤ ≤ ( 1 + C1(a, b, d, U, µκ(ξ), q) n ) EΥq k−1+ + C2(a, b, d, U, µκ(ξ), q) q−1∑ l=0 ( 1 + 1 n )l 1 nq−l EΥl k−1. (4.13) Let us show that (4.13) provide the family of estimates EΥq k ≤ C(a, b, d, U, µκ(ξ), q) ( k n )q , k = 1, . . . , n, q ∈ N (4.14) (note that (4.14) is exactly (4.5) with m = 1 and p = 2q). We use induction by q. For q = 1, (4.13) implies that EΥk ≤ C2(a, b, d, U, µκ(ξ), 1) n + + C2(a, b, d, U, µκ(ξ), 1) n ( 1 + C1(a, b, d, U, µκ(ξ), 1) n ) + . . . . . .+ C2(a, b, d, U, µκ(ξ), 1) n ( 1 + C1(a, b, d, U, µκ(ξ), 1) n )k−1 ≤ ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 369 ≤ k n C2(a, b, d, U, µκ(ξ), 1)eC1(a,b,d,U,µκ(ξ),1). Similarly, if (4.14) holds true for all q ≤ Q− 1, then (4.13) implies that EΥQ k ≤ C2(a, b, d, U, µκ(ξ), Q)eC1(a,b,d,U,µκ(ξ),Q)× × Q−1∑ l=0 2l nQ−l C(a, b, d, U, µκ(ξ), l) ( k n )l ≤ ≤ C(a, b, d, U, µκ(ξ), Q) ( k n )Q , that proves (4.14) for q = Q. This proves the statement of the lemma for m = 1. For arbitrary m, the proof is analogous: one should write difference relations for the higher derivatives of Xn, analogous to (4.3), and then again use the moment estimates of the same type with the given above. This step does not differ principally from the one for SDE’s driven by a Wiener process (see, for instance [13], Chapter V, § 8), and thus we omit its detailed exposition here. The lemma is proved. Corollary 4.1. The following estimate holds: Nj(Xn(t)) ≤ C(a, b, d, U, µκ(ξ), j) √ t, t ∈ [ 1 n , 1 ] , j ∈ N. Let us proceed with the investigation of the Malliavin’s matrix σfn for fn = Xn(t). Denote by En i,j , 0 ≤ i ≤ j ≤ n, the difference analogue of the stochastic exponent for (4.3), i.e., the family of Rd×d-valued variables satisfying the relations En i,i = IRd , En i,j = En i,j−1 +∇a ( Xn ( j − 1 n )) En i,j−1 1 n + + d∑ r=1 [ ∇br ( Xn ( j − 1 n )) En i,j−1 ξjr√ n ] , j = i, . . . , n. (4.15) Then one can easily obtain the representation for Yn(·), Yn ( k n ) = θn(ζ) k∑ j=1 d∑ r=1 1Iεj=1ψ(ηk) √ n [ En j,kbr ( Xn ( j − 1 n ))] ⊗ ejr. (4.16) Denote fn,k = X ( k n ) and σn,k = σfn,k . By (4.16), we have σn,k = θn(ζ) n k∑ j=1 d∑ r=1 [ψ2(ηj)1Iεj=1]× × [ En j,kbr ( Xn ( j − 1 n ))] ⊗ [ En j,kbr ( Xn ( j − 1 n ))] = ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 370 A. M. KULIK = θn(ζ) n k∑ j=1 [ψ2(ηj)1Iεj=1] [ En j,kb ( Xn ( j − 1 n ))][ En j,kb ( Xn ( j − 1 n ))]∗ = = θn(ζ) n k∑ j=1 [ψ2(ηj)1Iεj=1]En j,k[bb∗] ( Xn ( j − 1 n )) [En j,k]∗. Together with the family {En i,j}, we consider the family {Ẽn i,j} defined by Ẽn i,i = IRd , Ẽn i,j = Ẽn i,j−1 +∇a ( Xn ( j − 1 n )) Ẽn i,j−1 1 n + + d∑ r=1 [ ∇br ( Xn ( j − 1 n )) Ẽn i,j−1 ξn jr√ n ] , j = i, . . . , n, (4.17) where ξn i = ξi1I‖ζi‖≤nς , i = 1, . . . , n. By the construction, Ẽn i,j = En i,j on the set {θn(ζ) = 1}, therefore σn,k = θn(ζ) n k∑ j=1 [ψ2(ηj)1Iεj=1]Ẽn j,k[bb∗] ( Xn ( j − 1 n )) [Ẽn j,k]∗. We have Ẽn i,j = i+1∏ l=j [ IRd +∇a ( Xn ( l − 1 n )) 1 n + d∑ r=1 ∇br ( Xn ( l − 1 n )) ξn lr√ n ] . Since ∇a, ∇b are bounded and \|ξn jr\|1I‖ζj‖≤nς ≤ max(maxx∈U ‖x‖, nς), there exists n0 = n0(a, b, d, U, ς) such that∥∥∥∥∥∇a ( Xn ( l − 1 n )) 1 n + d∑ r=1 ∇br ( Xn ( l − 1 n )) ξn lr√ n ∥∥∥∥∥ 1I‖ζj‖≤nς ≤ 1 2 , n ≥ n0. Then Ẽn i,j is invertible and [Ẽn i,j ] −1 = j∏ l=i+1 [ IRd +∇a ( Xn ( l − 1 n )) 1 n + d∑ r=1 ∇br ( Xn ( l − 1 n )) ξn lr√ n ]−1 . Thus, on the set {θn(ζ) = 1}\{ε1 = . . . = εn = 0}, the matrix σn,k is invertible and ∥∥σ−1 n,k ∥∥ = [ inf ‖v‖=1 (σn,kv, v) ]−1 ≤ ≤ β−1(b) [ max i≤j≤n ∥∥[Ẽn i,j ]−1∥∥]2 1 n k∑ j=1 ψ2(ηj)1Iεj=1  −1 . (4.18) Lemma 4.3. For every p ≥ 1, ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 371 E [ max i≤j≤n ∥∥[Ẽn i,j ]−1∥∥]p ≤ C(a, b, d, U, µκ(ξ), p), n ∈ N. Proof. Since [ Ẽn i,j ]−1 = Ẽn 0,i [ Ẽn 0,j ]−1 , it is enough to prove that E[max i≤n ‖Ẽn 0,i‖]p ≤ C(a, b, d, U, µκ(ξ), p), (4.19) Emax j≤n ∣∣ det Ẽn 0,j ∣∣−p ≤ C(a, b, d, U, µκ(ξ), p). (4.20) Let us prove inequality E [ max i≤n ‖Ẽn 0,i‖2 ]p ≤ C(a, b, d, U, µκ(ξ), p) (4.21) with ‖A‖2 ≡ √∑ lr A2 lr; this will provide inequality (4.19). We deduce from (4.17) that Zn i ≡ ‖Ẽn 0,i‖22 satisfy relations analogous to (4.6), i.e., Zn i = Zn i−1 + V 1,n i−1 1 n + ∑ r V 2,n i−1 ξn ir√ n + ∑ r1,r2 V 3,n i−1 ξn ir1 ξn ir1 n , i = 1, . . . , n, (4.22) with an {Fi}-adapted sequences V 1,n i , V 2,n i,· , V 3,n i,·,· such that∣∣V 1,n i ∣∣ ≤ C(a, b, d, U)(1 + Zn i ), ∣∣V 2,n i,r ∣∣ ≤ C(a, b, d, U)Zn i ,∣∣V 3,n i,r1,r2 ∣∣ ≤ C(a, b, d, U)Zn i . (4.23) Then the moment estimates analogous to those made in the proof of Lemma 4.2 provide that max i≤n (EZn i ) p 2 ≤ C(a, b, d, U, µκ(ξ), p). (4.24) Denote An j = Zn 0 + ∑j i=1 ∆An i , M n j = ∑j i=1 ∆Mn i with ∆Mn i = ∑ r V 2,n i−1 ξn ir − Eξn ir√ n + ∑ r1,r2 V 3,n i−1 ξn ir1 ξn ir2 − Eξn ir1 ξn ir2 n and ∆An i = Zn i − Zn i−1 −Mn i = V 1,n i−1 1 n + ∑ r V 2,n i−1 Eξn ir√ n + ∑ r1,r2 V 3,n i−1 Eξn ir1 ξn ir2 n . Then Zi = Ai +Mi. By (4.12) and (4.23), \|∆An i \| ≤ C(a, b, U, µκ(ξ)) n (1 + Zi), and therefore (4.24) provides that ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 372 A. M. KULIK Emax j≤n \|An j \| r 2 ≤ C(a, b, U, µκ(ξ), p). Similarly, Burkholder inequality together with (4.23) and (4.9) – (4.12) provides that Emax j≤n \|Mn j \| p 2 ≤ C(a, b, U, µκ(ξ), p), that proves (4.21), and therefore (4.19). On the set { ‖A‖2 ≤ 1 2 } ⊂ Rd×d, the function Φ: A 7→ ( det[IRd + A] )−1 can be represented in the form Φ(A) = 1 +Q(A) + ϑ(A), where Q is a polynomial of A with degQ ≤ κ and \|ϑ(A)\| ≤ C‖A‖κ+1 2 . We have∥∥∥∥∥∇a ( Xn ( l − 1 n )) 1 n + d∑ r=1 ∇br ( Xn ( l − 1 n )) ξn lr√ n ∥∥∥∥∥ κ+1 2 ≤ C(a, b, d, U) n . Therefore, (det Ẽn 0,j) −1 = (det Ẽn 0,j−1) −1 [ 1 +Qn j−1 ( ξn j1√ n , . . . , ξn jd√ n ) + ϑn i ] , where \|ϑn i \| ≤ C(a, b, d, U) n , ϑn i is Fj-measurable,Qn j−1 is a polynomial with degQn j−1 ≤ ≤ κ and its coefficients are Fj−1-measurable and bounded by some constant depen- ding on the coefficients a, b. Repeating the arguments used in the proof of (4.19) we obtain (4.20). The lemma is proved. Lemma 4.4. For every p ∈ N, c > 0, E  k∑ j=1 ψ2(ηj)1Iεj=1  −p 1I∑k j=1 εj≥ck ≤ C(c, ψ, p)k−p, k ≥ 2p+ 1 c . Remark 4.1. For arbitrary ψ ∈ C∞(Rd) with ψ = 0 on ∂U, the given above statement may fail. It is crucial for ψ to have non-zero normal derivative at (some part of) the boundary in order to provide (4.25) below to hold true. Proof. Since η and ε are independent, E  k∑ j=1 ψ2(ηj)1Iεj=1  −p 1I∑k j=1 εj≥ck ≤ E  ]ck[∑ j=1 ψ2(ηj)  −p , where ]x[df= min{n ∈ Z \| n ≥ x}. By the construction of the function ψ, P(ψ2(ηj) ≤ z) ∼ C(ψ) √ z, z → 0 + . (4.25) Therefore ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 373 P(ψ2(η1) + . . .+ ψ2(ηl) ≤ z) ∼ C(ψ, l)z l 2 , z → 0+, and E(ψ2(η1) + . . .+ ψ2(ηl))−p < +∞ (4.26) as soon as l > 2p. We put q = 2p+ 1, N = [ ]ck[ q ] and divide the set {1, . . . , ]ck[} on the blocks {1, . . . , q}, {q + 1, . . . , 2q}, . . . , {(N − 1)q + 1, . . . , Nq}, {Nq + 1, . . . , ]ck[} (the last block may be empty). We denote ϑi = iq∑ j=(i−1)q+1 ψ2(ηj), i = 1, . . . , N. We have E  ]ck[∑ j=1 ψ2(ηj)  −p ≤ E ( N∑ i=1 ϑi )−p = N−pE ( 1 N N∑ i=1 ϑi )−p . The function x 7→ x−p is convex on R+, and therefore E ( 1 N N∑ i=1 ϑi )−p ≤ E ( 1 N N∑ i=1 ϑ−p i ) = Eϑ−p 1 < +∞ (the last inequality follows from (4.26)). If k ≥ 2p+ 1 c , then ]ck[ q ≥ 1 and therefore N = [ ]ck[ q ] ≥ ]ck[ 2q ≥ c 4p+ 2 k. Thus E  ]ck[∑ j=1 ψ2(ηj)  −p ≤ C(ψ, p) ( c 4p+ 2 )−p k−p. The lemma is proved. Inequality (4.18), Lemmas 4.3 and 4.4 provide the following estimate. For a given c > 0 and t ∈ [0, 1], we put Ξ̃n = {θn(ζ) = 1} ⋂{∑[tn] j=1 εj ≥ c[tn] } . Corollary 4.2. For p ∈ N and [tn] > 2p+ 1 c , E ∥∥%fn,Ξ̃n ∥∥p M ≤ C(a, b, d, U, ψ, µκ(ξ), p)t−p. At last, let us give an estimates for the tail probabilities for fn. The following lemma is completely analogous to Lemma 4.2; the proof is omitted. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 374 A. M. KULIK Lemma 4.5. For every p ≥ 1, there exist constant C(a, b, d, U, µκ(ξ), p) such that E‖Xn(t)−X(0)‖p Rd1I{θn(ζ)=1} ≤ C(a, b, d, U, µκ(ξ), p)t p 2 , t ∈ [ 1 n , 1 ] . Corollary 4.3. For every p ≥ 1, there exists constant Cp, dependent on a, b, d, U, µκ(ξ), p, such that P ( ‖Xn(t)−Xn(0)‖ ≥ y, θn(ζ) = 1 ) ≤ Cp ( 1 + ‖y‖2 t )−p , n ∈ N, t ∈ [ 1 n , 1 ] . Remark 4.2. For ‖y‖2 t small, the latter inequality is trivial since P(·) ≤ 1. For ‖y‖2 t large, it comes from Chebyshev’s inequality. Lemma 4.6. There exist constants C1, C2, C3, dependent on a, b, U, d, µκ(ξ), such that, for every λ ∈ Rd with ‖λ‖ ≤ C1n 1 κ+1 , Ee(λ,Xn(t)−Xn(0))1I{θn(ζ)=1} ≤ C2e C3t‖λ‖2 , t ∈ [ 1 n , 1 ] , n ∈ N. (4.27) Proof. For a given λ, denote Zn(t) = e(λ,Xn(t)−Xn(0)). We have Zn(0) = 1. On the other hand,∣∣∣∣ ξkr√ n ∣∣∣∣ ≤ maxy∈U ‖y‖√ n + nςn− 1 2 = maxy∈U ‖y‖√ n + n− 1 κ+1 on the set {θn(ζ) = 1}. Thus there exists a constant C4 such that, for ‖λ‖ ≤ C1n 1 κ+1 ,∣∣∣∣∣ 1n ( λ, a ( Xn ( k − 1 n ))) + d∑ r=1 ξkr√ n ( λ, br ( Xn ( k − 1 n )))∣∣∣∣∣ ≤ C4 on the set {θn(ζ) = 1}. Using the elementary inequality ex ≤ 1 + x + Cx2, \|x\| ≤ C4, we obtain that, on the same set, Zn ( k n ) = Zn ( k − 1 n ) exp [ 1 n ( λ, a ( Xn ( k − 1 n ))) + + d∑ r=1 ξkr√ n ( λ, br ( Xn ( k − 1 n )))] ≤ ≤ Zn ( k − 1 n )[ 1 + Θk−1 ‖λ‖+ ‖λ‖2 n + d∑ r=1 Λk−1,r ξkr(‖λ‖+ ‖λ‖2)√ n + + d∑ r1,r2=1 ∆k−1,r1,r2 ξkr1ξkr2‖λ‖2 n ] (4.28) with an Fk−1-measurable coefficients Θk−1, Λk−1,r, ∆k−1,r1,r2 , bounded by some constant C. An arguments, analogous to those used in the proof of Lemma 4.2, provide that (4.28) implies the estimate ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 375 EZn ( k n ) ≤ ( 1 + C ‖λ‖+ ‖λ‖2 n )k ≤ exp [ 2C k n (1 + λ2) ] , k, n ∈ N. (4.29) This is exactly (4.27) for t = k n . For t ∈ ( k n , k + 1 n ) , Zn(t) is a linear combination of Zn ( k n ) and Zn ( k + 1 n ) . Therefore, (4.27) follows from (4.29) and relation k + 1 n ≤ ≤ 2 k n ≤ 2t (recall that t ≥ 1 n and thus k ≥ 1). The lemma is proved. Corollary 4.4. There exist constants C5, C6, C7, dependent on a, b, U, d, µκ(ξ), such that P ( ‖Xn(t)−Xn(0)‖ ≥ y, θn(ζ) = 1 ) ≤ C6e −C7 y2 t , y ∈ (0,C5tn 1 κ+1 ), n ∈ N, t ∈ [ 1 n , 1 ] , and P ( ‖Xn(t)−Xn(0)‖ ≥ y, θn(ζ) = 1 ) ≤ C6e −C7n 1 κ+1 y, y ≥ C5tn 1 κ+1 , n ∈ N, t ∈ [ 1 n , 1 ] . Proof. It is enough to verify that, for any coordinate (Xn)j of the process Xn, j = 1, . . . , d, there exist constants C̃5, C̃6, C̃7, C̃8 such that P ( ± ((Xn)j(t)− (Xn)j(0)) ≥ y, θn(ζ) = 1 ) ≤ C̃6e −C̃7 y2 t , (4.30) y ∈ (0, C̃5tn 1 κ+1 ), n ∈ N, t ∈ [ 1 n , 1 ] , and P ( ± ((Xn)j(t)− (Xn)j(0)) ≥ y, θn(ζ) = 1 ) ≤ C̃6e −C̃8n 1 κ+1 y, (4.31) y ≥ C̃5tn 1 κ+1 , n ∈ N, t ∈ [ 1 n , 1 ] . Inequality (4.30) with C̃5 = 2C1C3 and C̃7 = [2C3]−1 follows from (4.27) with λ = = ( ± y 2C3t ) ej , where ej is the j-th coordinate vector in Rd. Inequality (4.31) with the same C̃5 and C̃8 = C1 2 follows from (4.27) with λ = ( ± C1n 1 κ+1 ) ej . Proof of Theorem 1.1. We take p = 8(d+ 1) and fix some c ∈ (0, α) (α is given in condition (B3)). We write n∗ = n0(a, b, U, ς) (see the notation before Lemma 4.3) and put ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 376 A. M. KULIK Ξt n =  {θn(ζ) = 1} ⋂ [tn]∑ j=1 εj ≥ c[tn] , n ≥ n∗, [tn] > 2p+ 1 c , ∅, otherwise, Qn x,t(dy) = P(Xn(x, t) ∈ dy,Ξt n) = PΞt n (fn x,t ∈ dy − x), Rn x,t(dy) = P(Xn(x, t) ∈ dy,Ω\Ξt n). Corollaries 4.1 and 4.2 provide condition (C1) of the Theorem 3.1. By the statement (a) of this theorem, Qn x,t(dy) = qn x,t(y) dy with qn x,t ≤ Kd(fn,Ξn)P 1 2 (‖fn‖ ≥ ‖y‖, θn(ζ) = 1). (4.32) Moreover, Corollaries 4.1 and 4.2 provide an explicit estimate for Kd(fn,Ξn). Namely, for some constant C dependent on a, b, c, d, µκ(ξ), U, ψ, Kd(fn,Ξn) ≤ C d∑ M=0 [ √ t]d+2M [t−4(M+d)] 1 4 = (d+ 1)C t− d 2 . (4.33) Thus the statement (a) of Theorem 3.1 and Corollaries 4.4, 4.3 provide statement (ii) of Theorem 1.1. By Chebyshev inequality, P(θn(ζ) = 0) ≤ nn−κ·ς = n−ε(κ). Take λ = ln ( α(1− c) c(1− α) ) > 0. By Chebyshev inequality, we get, after some simple calculations, P  k∑ j=1 εj < ck  ≤ Ee−λ ∑k j=1 εj e−λck = [Ψ(α, c)]k, k ∈ N, (4.34) with Ψ(α, c) = ( 1− α 1− c )1−c (α c )c . One can verify that Ψ(α, c) < 1 for 0 < c < α < < 1. Thus, we can conclude that, for ρ = −1 2 lnΨ(α, c) > 0, P(Ω\Ξt n) ≤ n−ε(κ) + e−ρnt when n ≥ n∗, [tn] > 2p+ 1 c( we have used here that [nt] ≥ nt 2 for t ≥ 1 n ) . Thus, for all t > 0, P(Ω\Ξt n) ≤ D [ n−ε(κ) + e−ρnt ] (4.35) with the constantD dependent on n∗, p, c, ρ. This provides statement (iii) of Theorem 1.1. We have shown that if x ∈ Rd, t > 0 are fixed then the functions fn = fn x,t and the sets Ξn = Ξt n satisfy all the conditions of Theorem 3.1. This means that ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 377 qn x,t(y) → px,t(y) uniformly w.r.t. y ∈ Rd. In order to show that this convergence holds uniformly w.r.t. t ∈ [δ, 1], x, y ∈ Rd, we need to show that, for every sequences {tn} ⊂ [δ, 1], {xn}, {yn} ⊂ Rd, qn xn,tn (yn)− pxn,tn (yn) → 0, n→∞. (4.36) We can suppose that {tn} converges to some t ∈ [δ, 1]. The functions a, b are bounded together with their derivatives up to the second order, and therefore the sequences of the functions an(·) = a(·+ xn), bn(·) = b(·+ xn) are pre-compact in C1(Rd,Rd) and C1(Rd,Rd×d), correspondingly. We can suppose that an → ã in C1(Rd,Rd), bn → b̃ in C1(Rd,Rd×d). Consider the processes Zn defined by the relations of the type (0.2), (0.3) with Zn(0) = 0 and the coefficients a, b replaced by an, bn. Also, consider the processes Zn defined by the stochastic differential equations of the type (0.1) with Zn(0) = 0 and the coefficients a, b replaced by an, bn. At last, consider the process Z defined by the stochastic differential equations of the type (0.1) with Z(0) = 0 and the coefficients a, b replaced by ã, b̃. Denote fn = Zn(tn), Ξn = Ξtn n . It is easy to verify that Zn converge weakly in C([0, 1],Rd) to Z (see, for instance, Proposition 5.1 [14]). Thus, for the sequences fn,Ξn, all the conditions of Theorem 3.1 hold true with f = Z(t). This means that f possesses a distribution density pf and sup y ∣∣pfn Ξn (y)− pf (y) ∣∣→ 0. (4.37) Similarly, one can show that, for fn = Zn(tn), the distribution density pfn exists and sup y ∣∣pfn (y)− pf (y) ∣∣→ 0. (4.38) Now, (4.36) is provided by the relations (4.37), (4.38) and qxn,tn(yn) = pfn Ξn (yn − xn), pxn,tn(yn) = pfn (yn − xn). This proves statement (i) of Theorem 1.1. The proof of (ii′) and (iii′) can be conducted analogously, with an appropriate changes of the truncation procedure and corresponding estimates. Under (Bexp 2 ), we put, instead of (4.2), θn(ζ) = 1Imaxk≤n ‖ζk‖≤δ √ n. By the Chebyshev’s inequality, P(θn(ζ) = 0) ≤ ne−κ(δ √ n)2 ≤ e−ρ̃n with an appropriate ρ̃ > 0. This and the estimate (4.34) provide statement (iii′). ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 378 A. M. KULIK Under (Bexp 2 ) and the truncation level given above, the estimates (4.10) – (4.12) have their (simpler) analogues, and thus the statement of Lemma 4.2 holds. The constant δ in the definition of the truncation level can be made small enough to provide inequality∥∥∥∥∥∇a ( Xn ( l − 1 n )) 1 n + d∑ r=1 ∇br ( Xn ( l − 1 n )) ξn lr√ n ∥∥∥∥∥ 1I‖ζj‖≤δ √ n ≤ 1 2 , n ≥ n0, to hold with some n0 dependent on a, b, U. Then the statement of Lemma 4.3 holds true, also. Lemma 4.4 does not depend on the truncation procedure. Thus the Corollaries 4.1, 4.2 hold true and provide the principal estimate (4.33). Under condition (Bexp 2 ), E [ exp [ 1 n ( λ, a ( Xn ( k − 1 n ))) + + d∑ r=1 ξkr√ n ( λ, br ( Xn ( k − 1 n )))] ∣∣∣∣F k−1 n ] ≤ ≤ C̃1e C̃s2 ‖λ‖2 n a.s. for every λ ∈ Rd with some constants C̃1, C̃2 dependent on a, b, d, κ, E eκ‖ξk‖2 . Then the arguments analogous to those made in the proof of Lemma 4.2 provide that the estimate (4.27) holds true for every λ ∈ Rd. Consequently, the first inequality in Corollary 4.4 holds true for every y > 0. This inequality, the estimate (4.33) and Theorem 3.1 provide (iii′). This completes the proof of Theorem 1.1. 4.2. Proof of Theorem 1.2. The implication 2 ⇒ 1 is obvious. Let us first prove 2 under additional supposition (B3). We put, in the notation of Section 3, θn(ζ) ≡ 1, fn = 1√ n ∑n k=1 ξk, Ξn =   n∑ j=1 εj ≥ cn  , n ≥ n∗, ∅, otherwise, with n∗, c that will be defined later. Then Dfn = 1√ n n∑ k=1 d∑ r=1 ψ(ηk)1Iεk=1br ⊗ ekr, where br stands for the r-th coordinate vector in Rd (the proof is straightforward and omitted). Since ψ is bounded on U together with all its derivatives, this provides the estimates analogous to those given in Lemma 4.2. The statement of Lemma 4.3 is trivial now, since Ei,j = IRd (the identity matrix in Rd) for every i, j. Now, take p = 8(d+ 1), c ∈ (0, α), n∗ > 2p+ 1 c . Using Lemma 4.4, we obtain the estimate (4.33) with the ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 379 constant C dependent on c, d, ψ. The estimate (4.34) provides that P(Ξn) ≤ De−ρn for an appropriate D, ρ > 0. Now the statement 2 follows from Theorem 3.1. Let us replace the additional supposition (B3) by the condition 1. It is enough to prove the statement 2 for n ∈ mN with some given m ∈ N. We take m = 2n0 with n0 given in the statement 1 and have d[Pm]ac dλd ≥ d[Pn0 ] ac dλd ∗ d[Pn0 ] ac dλd . The function d[Pn0 ] ac dλd ∗ d[Pn0 ] ac dλd is continuous since λd-almost all points of Rd are a Lebesgue points (i.e., a points of λd-almost continuity) for any function g ∈ L1(Rd). In addition, d[Pn0 ] ac dλd ∗ d[Pn0 ] ac dλd is not an identical zero due to the statement 1. Thus there exist α̃ > 0 and an open set U ⊂ Rd such that d[Pn0 ] ac dλd ∗ d[Pn0 ] ac dλd ≥ α̃1IU . Therefore the distribution of ξ1 + . . .+ ξm satisfies (B3). Using what we have proved before, we deduce that the statement 2 holds for n ∈ mN, and therefore for n ∈ N. This completes the proof of Theorem 1.2. 4.3. Sketch of the proof of Theorem 2.1. We will show that, under conditions of Theorem 2.1, the following uniform local Doeblin condition holds true. For two measures µ1, µ2, denote [µ1 ∧ µ2](dy) = min [ dµ1 d(µ1 + µ2) (y), dµ2 d(µ1 + µ2) (y) ] (µ1 + µ2)(dy). Proposition 4.1. For every ball B there exists nB ∈ N, TB , γB > 0 such that[ Px,TB ∧ Px′,TB ] (Rd) ≥ γB , x, x′ ∈ B, (4.39) and, for every n ≥ nB , there exists Tn B ∈ 1 n Z+, T n B ≤ TB such that [ Pn x,T n B ∧ Pn x′,T n B ] (Rd) ≥ γB , x, x′ ∈ B, n ≥ nB . (4.40) Once Proposition 4.1 is proved, one can finish the proof of Theorem 2.1 following the proof of Theorem 1 in [4] literally. We omit this part of the discussion and prove Proposition 4.1, only. Proof of Proposition 4.1. Since (B5) implies (Bκ 2 ) for any κ, we can apply Theorem 1.1. One can easily see that[ Pn x,t ∧ Pn x′,t ] (Rd) ≥ ∫ Rd min[qn x,t(y), q n x′,t(y)] dy. Thus, for any sequence tn → t > 0 we have, by the statement (i), lim inf n→+∞ inf x,x′∈B [ Pn x,t ∧ Pn x′,t ] (Rd) ≥ inf x,x′∈B ∫ Rd min [ px,t(y), px′,t(y) ] dy. On the other hand (see [2]), under condition (B1) the function ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 380 A. M. KULIK Rd × (0,+∞)× Rd 3 (x, t, y) 7→ px,t(y) is continuous and strictly positive at every point. Therefore, for a given B, γB df= 1 2 inf x,x′∈B ∫ Rd min [ px,t(y), px′,t(y) ] dy > 0, and (4.39), (4.40) hold true for TB = Tn B = 1 and sufficiently large nB . The proposition is proved. 4.4. Sketch of the proof of Theorem 2.2. It was already mentioned in Subsecti- on 2.1 that Theorem 2.2 is analogous to Theorem 2.1 [5]. We refer the reader to the paper [5] for the detailed proof. Here, we expose a principal estimate only, demonstrati- ng that, in this proof, the truncated local limit theorem can be used efficiently instead of the usual one, that was used in [5]. Theorem 2.1 [5] is derived from the general theorem on convergence in distribution of a sequence of additive functionals of Markov chains, given in the paper [11] (Theorem 1). The characteristics of the functionals ϕ(X), ϕn(Xn) are defined by the relations f t(x) df= E [ ϕ0,t(Xn) \| X(0) = x ] , fs,t n (x) df= E [ ϕs,t n (Xn) \| Xn(s) = x ] , s = i n , i ∈ Z+, t > s, x ∈ Rd. The first relation is due to [10], Chapter 6. The second relation was introduced in [11] by an analogy with the first one. The key condition of Theorem 1 [11] is sup x∈Rd, s= i n , t∈(s,T ) ∣∣fs,t n (x)− f t−s(x) ∣∣→ 0, n→∞. (4.41) Here, we need to verify this condition only, since, for all the other conditions, the proof from [5] can be used literally. We have fs,t n (x) = 1 n Fn(x) + 1 n ∑ k∈N, k n <t−s ∫ Rm Fn(y)Pn x, k n (dy) = f0,t−s n (x), s ≤ t, x ∈ Rm. We use the decomposition Pn = Qn +Rn from Theorem 1.1 and write f0,t n (x) = 1 n Fn(x) + 1 n ∑ k∈N, k n <t ∫ Rm Fn(y)Rn x, k n (dy) + 1 n ∑ k∈N, k n <t ∫ Rm Fn(y)qn x, k n (y)dy. The statement (ii) of Theorem 1.1 and the estimates, analogous to the estimates (4.2) – (4.10) from [5], imply that sup x∈Rd,t≤T ∣∣∣∣∣∣ 1n ∑ k∈N, k n <t ∫ Rm Fn(y)qn x, k n (y) dy − f t(x) ∣∣∣∣∣∣→ 0, n→∞. On the other hand, ε(κ) = 8 7 > 1 for κ = 6. Thus, by condition (B6) and the statement (iii) of Theorem 1.1, ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3 MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL ... 381 E  1 n Fn(x) + 1 n ∑ k∈N, k n <t ∫ Rm Fn(y)Rn x, k n (dy)  ≤ ≤ n−1 sup x′ Fn(x′) [ 1 + ∑ k<tn Rn x, k n (Rd) ] ≤ ≤ n−1 sup x′ Fn(x′) [ 1 +Dn− 8 7nt+D ∑ k∈N e−γk ] → 0, n→∞, uniformly for x ∈ Rd, t ≤ T for any T ∈ R+. This proves (4.41). 1. Skorokhod A. V. Asymptotic methods in the theory of stochastic differential equations. – Kiev: Naukova dumka, 1987. – 328 р. 2. Il’in A. M., Kalashnikov A. S., Oleinik O. A. Linear second order parabolic equations // Uspekhi Mat. Nauk. – 1962. – 17, № 3. – P. 3 – 143. 3. Konakov V., Mammen E. Local limit theorems for transition densities of Markov chains converging to diffusions // Probab.Theory and Relat. Fields. – 2000. – 117. – P. 551 – 587. 4. Klokov S. A., Veretennikov A. Yu. Mixing and convergence rates for a family of Markov processes approximating SDEs // Random Oper. and Stochast. Equat. – 2006. – 14, № 2. – P. 103 – 126. 5. Kulik A. M. Difference approximation for local times of multidimensional diffusions. – Preprint is available at arxiv:math.PR/0702175. 6. Konakov V. Small time asymptotics in local limit theorems for Markov chains converging to diffusions. – 2006. – arxiv:math. PR/0602429. 7. Ibragimov I. A., Linnik V. Yu. Independent and stationary connected variables. – Moscow: Nauka, 1965. – 524 p. 8. Veretennikov A. Yu. On estimates of mixing rate for stochastic equations // Teor. Veroyatnost. i Prim. — 1987. – 32. – P. 299 – 308. 9. Veretennikov A. Yu. On polynomial mixing and rate of convergence for stochastic differential and difference equations // Teor. Veroyatnost. i Prim. – 1999. – 44. – P. 312 – 327. 10. Dynkin E. B. Markov processes. – Moscow: Fizmatgiz, 1963. – 860 p. 11. Kartashov Yu. N., Kulik A. M. Invariance principle for additive functionals of Markov chains. – Preprint is available at arXiv:0704.0508v1. 12. Nualart D. Analysis on Wiener space and anticipating stochastic calculus // Lect. Notes Math. – 1998. – 1690. – P. 123 – 227. 13. Ikeda N., Watanabe S. Stochastic differential equations and diffusion processes. – Amsterdam: North-Holland, 1981. – 456 p. 14. Kurtz T. G., Protter Ph. Weak limit theorems for stochastic integrals and SDE’s // Ann. Probab. – 1991. – 19, № 3. – P. 1035 – 1070. Received 10.01.08 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 3

Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem

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