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 Lp 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.

A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations

“…[2] obtained the difference between approximate solution and accurate solution for the stochastic differential equations where the approximate solution is called Caratheodory's approximate solution which has been constructed by successive approximations. [3] presented result on an analytic approximate method for the class of stochastic differential equations with coefficients that do not satisfy the Lipschitz and linear growth conditions but behaved like a polynomial. Furthermore, equations from this class have unique solutions with bounded moments and their coefficients satisfy polynomial conditions.…”