Convergence and stability of exponential integrators for semi-linear stochastic variable delay integro-differential equations

ABSTRACT

In this paper, the numerical method for semi-linear stochastic variable delay integro-differential equations is studied. The stability of analytic solutions of semi-linear stochastic variable delay integro-differential equations are studied first, some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential integrators for semi-linear stochastic variable delay integro-differential equations are constructed, the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with the strong order 1/2 and the exponential Euler method can keep the mean-square exponential stability of the analytical solution under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

Keywords:

• Semi-linear stochastic variable delay integro-differential equation
• exponential integrators
• mean-square exponential stability
• convergence
• strong convergence order

2010 Mathematics Subject Classifications:

• 60H35
• 65C20

Acknowledgments

The authors would like to thank the referees for giving useful suggestions and comments for the improvement of this paper.

Disclosure statement

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

Additional information

Funding

This research is supported by National Natural Science Foundation of China [Project No. 11901173] and by the Heilongjiang Provincial Natural Science Fund [LH2019A030] and by the Innovative team projects of Heilongjiang institute of technology [2018CX17].