Donsker’s delta functions and approximation of heat kernels by the time discretization methods
Donsker’s delta functions and approximation of heat kernels by the time discretization methods
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Donsker 的 delta 函数和时间离散方法的热核近似
DOI：
10.1215/kjm/1250518506
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发表时间：
1996
影响因子：
--
通讯作者：
Shinzo Watanabe
中科院分区：
文献类型：
--
作者：
Yaozhong Hu;Shinzo Watanabe
随机微分方程解的时间离散近似格式已经被许多人研究过，例如，在Kloeden-Platen的书[K1-P192]。由于热核是解的定律的概率密度，因此，如果要问这些近似方案是否提供了热核的自然近似方案，可能是不必要的。本文的目的是提出一个这样的计划与Malliavin演算的帮助。在第一节中，我们引入了Donsker δ函数的概念，它是Wiener空间上的一类广义Wiener泛函。在第二节中，我们得到了Donsker de lta函数的一个一般逼近结果。在第三节中，我们考虑随机微分方程解所给出的Wiener泛函的情形。Kloeden和Platen [K1-P195]曾提出一个y阶的Iti 5-Taylor逼近格式。本文将他们关于Malliavin积分的L2 -模强收敛的结果改进为关于每一Sobolev模的强收敛（定理3.1）.这是本文的一个主要结果，并在第四节中给出了证明。这个结果与第2节中的一般结果相结合，得到了Donsker δ函数的强逼近方案，从而得到了热核的定理3.2形式的逼近结果.然而，应该注意的是，热核是由Donsker δ函数的广义期望给出的，因此，这个问题所涉及的本质上是一个弱近似。定理3.2的收敛率是强逼近的收敛率，可以改进为弱收敛的收敛率。对于这样的改进，我们参考了Bally和Talay [B-T95]和Kohatsu-Higa [Ko 95]的近期作品。
Time discretization approximation schemes for solutions of stochastic differential equations have been studied by m any people and are treated, e.g., in the book of Kloeden-Platen [K 1-P192]. Since heat kernels a re the probability densities of the law of solutions, it might be worth-while to ask if these approximation schemes provide a natural scheme of approximation for heat kernels. Purpose of this paper is to propose one of such schemes with a help of Malliavin calculus. In section 1, we introduce the notion of Donsker's delta functions a s a class of generalized Wiener functionals on W iener sp a c e . In section 2 , w e obtain a general approximation result fo r Donsker's de lta functions. In sec tion 3 , we consider the case of Wiener functionals given by solutions to stochastic differential equations. A n Iti5-Taylor approximation scheme of order y for the solution has been introduced by Kloeden and Platen [K l-P195]. Here we improve their result o f the strong convergence in the L2 -norm t o th e strong convergence in every Sobolev norm in the Malliavin calculus (Theorem 3.1). This is a m ain result of this paper and its proof is given in section 4. This result, combined with general results in section 2, yields some strong approximation scheme for Donsker's delta functions a n d thereby an approxim ation result fo r the heat kernel in th e form of Theorem 3.2. However, it should be remarked that the heat kernel is given by a generalized expectation o f Donsker's delta function a n d therefore, what is involved in this problem is essentially a n w eak approxim ation . T he ra te of convergence in Theorem 3.2 is th a t o f the strong approximation and it can be im proved to th e ra te o f weak convergence. For such improvements, we refer to the recent works by Bally and Talay [B-T95] and Kohatsu-Higa [Ko95].