Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay

Abstract

This article can be considered as a continuation of Petrović and Milošević [The truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay, Filomat 35 (2021), no. 7, 2457–2484], where the authors established the L q {L}^{q} -convergence of the truncated Euler-Maruyama (EM) method for neutral stochastic differential equations with time-dependent delay under the Khasminskii-type condition. However, the convergence rate of the method has not been studied there, which is the main goal of this article. Also, there are some restrictions on the truncated coefficients of the considered equations, and these restrictions sometimes might force the step size to be so small that the application of the truncated EM method would be limited. Therefore, the convergence rate without these restrictions will be considered in this article. Moreover, one of the sufficient conditions for obtaining the main result of this article, which is related to Lipschitz constants for the neutral term and delay function, is weakened. In that way, some of the results of the cited article are generalized. The main result of this article is proved by employing two conditions related to the increments to the coefficients and the neutral term of the equations under consideration, among other conditions. The main theoretical result is illustrated by an example.