Strong convergence of the tamed Euler method for nonlinear hybrid stochastic differential equations with piecewise continuous arguments

This paper develops the tamed Euler method (TEM) for approximating solutions of nonlinear hybrid stochastic differential equations with piecewise continuous arguments (SDEPCAs). As far as we know, there are only two published articles (Hou et al., 2009; Zhang, 2020) considered the Lp-convergence rate of numerical methods for hybrid systems, and the rates are no more than 1/p (p > 2) in both of these two studies. In this work, we use the technique proposed in Song et al. (2022) to develop the TEM, and obtain the following two main results: (i) The TEM converges strongly to the exact solution of hybrid SDEPCAs; (ii) The Lp-convergence rate (p > 2) can reach 1/2. Finally, we give two numerical examples to verify the theoretical results. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper develops the tamed Euler method (TEM) for approximating solutions of nonlinear hybrid stochastic differential equations with piecewise continuous arguments (SDEPCAs). The TEM converges strongly to the exact solution of hybrid SDEPCAs, and the Lp-convergence rate (p > 2) can reach 1/2. Two numerical examples are provided to verify the theoretical results.