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Zbl 1115.65007
Ding, Xiaohua; Wu, Kaining; Liu, Mingzhu
Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. (English)
[J] Int. J. Comput. Math. 83, No. 10, 753-761 (2006). ISSN 0020-7160; ISSN 1029-0265/e

As explicit solutions can rarely be obtained for stochastic delay integro-differential equations (SDIDEs), it is useful to develop numerical approximations, even in the linear case. It is proved in this paper that a semi-implicit Euler method is convergent with strong order $p=0.5$. Under a simple condition on the coefficients, the Lyapunov function method proves that the zero solution of the SDIDE is asymptotic mean square stable. The same is true for the semi-implicit Euler method with suitable time step. Numerical experiments are presented.
[Dominique Lepingle (Orléans)]

MSC 2000:: *65C30 Stochastic differential and integral equations
60H20 Stochastic integral equations
60H35 Computational methods for stochastic equations
45R05 Random integral equations
65R20 Integral equations (numerical methods)

Keywords: stochastic delay integro-differential equations; mean square stability; semi-implicit Euler method; numerical solution; Lyapunov function method; numerical experiments

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Zentralblatt MATH Berlin [Germany]