Continuous Weak Approximation for Stochastic Differential Equations

TL;DR

This paper extends Milstein's convergence theorem to continuous weak approximations of stochastic differential equations, providing second order conditions for stochastic Runge-Kutta methods and optimal coefficients for multi-dimensional cases.

Contribution

It generalizes a key convergence theorem and derives new second order conditions and optimal coefficients for continuous stochastic Runge-Kutta methods.

Findings

Proved a convergence theorem extending Milstein's work.

Derived uniform second order conditions for stochastic Runge-Kutta methods.

Presented coefficients for optimal schemes for multi-dimensional Wiener processes.

Abstract

A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations by general one step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge-Kutta methods containing the continuous extension of the second order stochastic Runge-Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to It\^o stochastic differential equations with respect to a multi-dimensional Wiener process are presented.