On the approximations of solutions to stochastic differential equations under polynomial condition

Abstract

The subject of this paper is an analytic approximate method for a class of stochastic differential equations with coefficients that do not necessarily satisfy the Lipschitz and linear growth conditions but behave like a polynomials. More precisely, equations from the observed class have unique solutions with bounded moments and their coefficients satisfy polynomial condition. Approximate equations are defined on partitions of a time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. The rate of $L^p$ convergence increases when degrees in Taylor approximations of coefficients increase. At the end of the paper, an example is provided to support the main theoretical result.