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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:
65C30Stochastic differential and integral equations
60H20Stochastic integral equations
60H35Computational methods for stochastic equations
45R05Random integral equations
65R20Integral equations (numerical methods)