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Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. (English) Zbl 1115.65007
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.
##### MSC:
 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)