The truncated Euler–Maruyama method for stochastic differential equations with piecewise continuous arguments driven by Lévy noise

ABSTRACT

This paper aims to consider stochastic differential equations with piecewise continuous arguments (SDEPCAs) driven by Lévy noise where both drift and diffusion coefficients satisfy local Lipschitz condition plus Khasminskii-type condition and the jump coefficient grows linearly. We present the explicit truncated Euler–Maruyama method. We study its moment boundedness and its strong convergence. Moreover, the convergence rate is shown to be close to that of the classical Euler method under additional conditions.

KEYWORDS:

• Stochastic differential equation
• truncated Euler–Maruyama method
• local Lipschitz condition
• Khasminskii-type condition
• convergence rate

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65L05
• 65L60

Acknowledgments

The authors wish to thank the referees for their helpful comments which improve this paper significantly. The authors would also like to thank Prof. Xuerong Mao's help. This work is supported by Basic Scientific Research in Colleges and Universities of Heilongjiang Province (special fund project of Heilongjiang university) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems.

Disclosure statement

No potential conflict of interest was reported by the author.