Convergence and stability of modified partially truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments

Abstract

AbstractThis paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.Keywords: Modified partially truncated EM methodstochastic differential equations with piecewise continuous argumentsstrong convergenceconvergence ratemean square exponential stabilityDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is supported by the NSF of PR China (No. 12071101 and No. 11671113).