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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 $$ dx(t)=f(x_{t})dt+g(x_{t})dB_{t}+h(x_{t})dN(t),\ 0\leq t\leq T, $$ with given $x_{0}$, where $x$ is $n$-dimensional,$\ x_{t}:=\{x(t+\theta ),\ -\tau\leq\theta\leq0\}$, $x_{t^{-}}:=\{x((t+\theta)^{-}),\ -\tau\leq \theta\leq0\}$,$\ \ \ x(t^{-}):=\lim_{s\uparrow t}x(s)$, $B(t)$ is an $m$-dimensional Brownian motion, and $N(t)$ is a scalar Poisson process. Under a global Lipschitz condition they show that the $p$th-moment convergence of Euler-Maruyama numerical solutions to SFDEs with jumps has the order $1/p$ for any $p\geq2$. This is different from the case of SFDEs without jumps, where the order is $1/2$ for any $p\geq2$. They consider also the mean-square convergence under a local Lipschitz condition.

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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