A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditions

Abstract

In this paper, we aim to study strong convergence of theta-EM scheme for stochastic differential equations under a local one-sided Lipschitz condition. Since the existence and uniqueness of theta-EM solution cannot be guaranteed under the local one-sided Lipschitz condition, a modified theta-EM scheme is proposed, and strong convergence of the numerical solution to the exact solution is given.

Keywords:

• Stochastic differential equations
• implicit scheme
• local one-sided Lipschitz
• strong convergence
• discrete Gronwall inequality

AMS Subject Classifications:

• 65C30
• 65L20

Acknowledgments

The authors would like to thank the associated editor and referees for their helpful comments and suggestions. They would also like to thank Huabin Chen and Ziheng Chen for their useful suggestions.

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China (NSFC) [grant numbers 11561027, 11661039, 71563014, 71371193] and the NSF of Jiangxi [grant numbers 20192ACBL21007, 20181BAB201005, 2018ACB21001, GJJ150444].