A Taylor approximation of the solutions of neutral stochastic differential equations with constant delay under polynomial conditions

Abstract

Neutral stochastic differential equations with constant delay (NSDDEs) arise from describing many natural and social phenomena, such as those from medicine, physics, chemistry, epidemiology, neural science and economy. These equations are non-solvable in most cases. In this paper that class of equations is analyzed under the polynomial conditions on their coefficients. Approximate solution to initial NSDDE is obtained as the solution of the equation acquired by the Taylor expansion with respect to the delayed argument of the coefficients of that equation. The existence and uniqueness of newly formed approximate equation is shown and the closeness between solutions of the approximate and initial equations is estimated in the $L^p$ and almost sure sense. The rate of convergence increases as the orders of the Taylor expansions of the coefficients increase. Theoretical results are derived from various elementary, integral, probability and martingale-related inequalities. Examples are provided in order to support theoretical results.