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Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise. (English) Zbl 1345.60051
The authors consider the problem of approximation of the solutions of stochastic integral equations with Lévy noise
$\begin{split} X_\varepsilon(t)= X(0)+ \varepsilon\int^t_0 b(s, X_\varepsilon(s-))\,ds+ \sqrt{\varepsilon} \int^t_0 \sigma(s,X_\varepsilon(s-))\,dB(s)\\+ \sqrt{\varepsilon}\int^t_0 \int_{|x|<c} F(s, X_\varepsilon(s-), x)\,\widetilde N(ds, dx)\end{split}$ by the solution of the corresponding solution of the simplified stochastic integral equations with Lévy noise
$\begin{split} Z_\varepsilon(t)= X(0)+ \varepsilon\int^t_0\overline b(Z_\varepsilon(s-))\,ds+ \sqrt{\varepsilon}\int^t_0\overline\sigma(Z_\varepsilon(s-))\,dB(s)\\ +\sqrt{\varepsilon} \int^t_0 \int_{|x|<c}\overline F(Z_\varepsilon(s-), x)\,\overline N(ds, dx),\end{split}$ where $$X(0)=\xi$$ is the initial value $$n$$-dimensional random vector with $$E[|\xi|^2]<\infty$$, $$b(t,\bullet)$$ and $$F(t,\bullet,x)$$ are given vector functions, $$\sigma(t,\bullet)$$ is an $$(n\times r)$$-matrix, $$b(t,y)$$, $$\sigma(t,y)$$ and $$F(t,y,x)$$ are continuous in $$y$$ for each fixed $$t\in[0,T]$$ and satisfy quadratic (non-Lipschitz) conditions for all $$y$$. Moreover, $$\overline b(y)$$, $$\overline\sigma(y)$$ and $$\overline F(y,x)$$ are measurable and satisfy, with the corresponding functions $$b(t,y)$$, $$\sigma(t, y)$$ and $$F(t,y,x)$$, a linear (Lipschitz) condition for the drift functions and quadratic conditions for the other functions. Also, $$\varepsilon\in(0,\varepsilon_0]$$ is a positive parameter with $$\varepsilon_0$$ a fixed number. The convergence of the solutions of these equations in mean-square and in probability is shown in this paper.
These results are extended to the case of stochastic functional integral equations with Lévy noise. Unfortunately, there are no examples showing the applicability of the obtained results.

MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 60H20 Stochastic integral equations 60G51 Processes with independent increments; Lévy processes 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 34A45 Theoretical approximation of solutions to ordinary differential equations 45R05 Random integral equations 34K50 Stochastic functional-differential equations
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