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Convergence rate of numerical solutions to SFDEs with jumps. (English) Zbl 1236.65005

The authors consider stochastic functional differential equations (SFDEs) with jumps of the form

dx(t)=f(x t )dt+g(x t )dB t +h(x t )dN(t),0tT,

with given x 0 , where x is n-dimensional,x t :={x(t+θ),-τθ0}, x t - :={x((t+θ) - ),-τθ0},x(t - ):=lim st x(s), B(t) is an m-dimensional Brownian motion, and N(t) is a scalar Poisson process.

Under a global Lipschitz condition they show that the pth-moment convergence of Euler-Maruyama numerical solutions to SFDEs with jumps has the order 1/p for any p2. This is different from the case of SFDEs without jumps, where the order is 1/2 for any p2. They consider also the mean-square convergence under a local Lipschitz condition.

MSC:
65C30Stochastic differential and integral equations
65L20Stability and convergence of numerical methods for ODE
60H35Computational methods for stochastic equations
34K50Stochastic functional-differential equations
34K28Numerical approximation of solutions of functional-differential equations
34F05ODE with randomness
60H10Stochastic ordinary differential equations
60J65Brownian motion
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