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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.

##### MSC:
 65C30 Stochastic differential and integral equations 65L20 Stability and convergence of numerical methods for ODE 60H35 Computational methods for stochastic equations 34K50 Stochastic functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations 34F05 ODE with randomness 60H10 Stochastic ordinary differential equations 60J65 Brownian motion
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