High order Itô-Taylor approximations to heat kernels
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.
scientific article; zbMATH DE number 1140498