High order Itô-Taylor approximations to heat kernels
type: Item
instance of: scholarly article
title: High order Itô-Taylor approximations to heat kernels
zbMATH Open document ID: 0901.60029
published in: Journal of Mathematics of Kyoto University
publication date: 1998-11-11T00:00:00Z
review text: Let \((\Omega, \mathcal F, P)\) be the canonical space of the standard \(m\)-dimensional Wiener process \(W= (W^1,\dots, W^m)\). On this space, define \(X\) as the solution of the following stochastic differential equation \[ X_t=x+\sum_{i=1}^m\int_0^tB_i(X_s)dW_s^i+ \int_0^tB_0(X_s)ds, \qquad 0\leq t\leq T, \] where \(x\in\mathbb{R}^d\) and \(B_i: \mathbb{R}^d\to\mathbb{R}^d, i=0,\dots, m,\) are smooth vector fields with bounded derivatives. The purpose of this article is to find a way of approximating a smooth density of \(X_t\). The author proves that the rate of convergence for this problem can be considered as the weak approximation rate. The existence of an expansion of the error in terms of the stepsize of the approximation is obtained. These results could have a variety of applications when coupled with Monte Carlo methods in order to simulate approximations of heat kernels with a potential.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1140498
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: Wiener process
zbMATH Keywords: Itô-Taylor approximation
zbMATH Keywords: convergence rate of the density
zbMATH Keywords: Euler-Maruyama scheme
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: Donsker delta functions
zbMATH Keywords: approximations of heat kernels
zbMATH Keywords: Monte Carlo methods
author: Arturo Kohatsu-Higa
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518401
OpenAlex ID: W1504696424
DOI: 10.1215/KJM/1250518401
description: scientific article; zbMATH DE number 1140498