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

ABSTRACT

This 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 the underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.

KEYWORDS:

• Modified partially truncated EM method
• stochastic differential equations with piecewise continuous arguments
• strong convergence
• convergence rate
• mean square exponential stability

2000 AMS SUBJECT CLASSIFICATION:

• 65C30
• 65H35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the NSF of PR China (No. 12071101 and No. 11671113).