CONVERGENCE AND MEAN-SQUARE STABILITY OF EXPONENTIAL EULER METHOD FOR SEMI-LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS

doi:10.4208/jcm.2010-m2019-0200

Journal of Computational Mathematics ›› 2022, Vol. 40 ›› Issue (2): 177-204.DOI: 10.4208/jcm.2010-m2019-0200

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CONVERGENCE AND MEAN-SQUARE STABILITY OF EXPONENTIAL EULER METHOD FOR SEMI-LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS

Haiyan Yuan

Department of Mathematics, Heilongjiang Institute of Technology, Harbin 158100, China

Received:2019-09-11 Revised:2020-04-26 Online:2022-03-15 Published:2022-03-29
Supported by:
This research is supported by National Natural Science Foundation of China (Project No.11901173) and by the Heilongjiang province Natural Science Foundation (LH2019A030) and by the Heilongjiang province Innovation Talent Foundation (2018CX17).

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Haiyan Yuan. CONVERGENCE AND MEAN-SQUARE STABILITY OF EXPONENTIAL EULER METHOD FOR SEMI-LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS[J]. Journal of Computational Mathematics, 2022, 40(2): 177-204.

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

Key words: Semi-linear stochastic delay integro-differential equation, Exponential Euler method, Mean-square exponential stability, Trapezoidal rule

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CONVERGENCE AND MEAN-SQUARE STABILITY OF EXPONENTIAL EULER METHOD FOR SEMI-LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS

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