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Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems. (English) Zbl 1144.60038
Consider two-time-scale systems of stochastic differential equations with Poisson jumps $dX^\varepsilon(t) = a(X^\varepsilon(t),Y^\varepsilon(t)) dt + b(X^\varepsilon(t))dB(t) + c(X^\varepsilon(t))dP(t), X^\varepsilon(0)=x_0 \in R^n,$ \begin{aligned} dY^\varepsilon(t) = \frac{1}{\varepsilon} f(X^\varepsilon(t), Y^\varepsilon(t)) dt + \frac{1}{\sqrt{\varepsilon}} g(X^\varepsilon(t),\\ Y^\varepsilon(t))dW(t) + h(X^\varepsilon(t),Y^\varepsilon(t))dN^\varepsilon(t), Y^\varepsilon(0)=y_0 \in R^m,\end{aligned} driven by multi-dimensional, independent Wiener processes $$B$$ and $$W$$, and $$P$$ and $$N^\varepsilon$$ scalar Poisson processes with certain intensities. The main goal is to study the strong convergence rate of the slow components $$X^\varepsilon$$ to the limit dynamics of $$\bar{X}$$, uniformly in time $$t$$. Under ergodicity assumptions which are more general than the one-sided Lipschitz condition, the authors use a Lyapunov function to control the return times. A strong limit theorem (in the strong $$L^2$$-sense, uniform in $$t$$) for the averaging principle of the form $E ( \sup_{0 \leq t \leq T} | X^\varepsilon(t)-\bar{X}(t)| ^2 \leq C (\varepsilon)$ with well-determined $$C(0+)=0$$ is proved for sufficiently small $$\varepsilon>0$$.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 60H30 Applications of stochastic analysis (to PDEs, etc.) 60J75 Jump processes (MSC2010) 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 65C30 Numerical solutions to stochastic differential and integral equations
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