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Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion. (English) Zbl 1365.34102
This paper deals with a stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion. The authors consider stochastic differential equations with fractional Brownian motion $X_{\varepsilon}(t)=X(0)+\varepsilon^{2H}\int_{0}^{t}b(s,X_{\varepsilon}(s))ds+ \varepsilon^{H}\int_{0}^{t}\sigma(s,X_{\varepsilon}(s))d^{-}B^{H}(s),\quad X(0)=X_0,$ where $$d^{-}B^{H}(s)$$ denotes the forward integral case, $$B^{H}(s)$$ is a fractional Brownian motion with Hurst index $$H\in(1/2,1)$$, the coefficients satisfy non-Lipschitz conditions, $$\varepsilon\in (0,\varepsilon_0]$$. Suppose that there exist functions $$\bar b(X),\bar\sigma(X)$$ such that ${1\over T_1}\int_0^{T_1}| b(s,X)-\bar b(X)| ds\leq \varphi_1(T_1)\rho(| X|), \;{1\over T_1}\int_0^{T_1}| \sigma(s,X)-\bar \sigma(X)|^2 ds\leq \varphi_2(T_1)\rho(| X|^2),$ where $$T_1\in[0,T]$$, $$\varphi_{i}$$ are positive bounded functions with $$\lim_{T_1\to\infty}\varphi_{i}(T_1)=0$$, $$i=1,2$$, $$\rho(\cdot)$$ is a non-decreasing, continuous and concave function from $$R^{+}$$ to $$R^{+}$$, and $$\bar b:R\to R$$, $$\bar\sigma:R\to R$$ are all measurable functions. The main result of paper is the following. If $$Z_{\varepsilon}(t)$$ denotes the solution of the stochastic differential equations $Z_{\varepsilon}(t)=X(0)+\varepsilon^{2H}\int_{0}^{t}\bar b(Z_{\varepsilon}(s))ds+ \varepsilon^{H}\int_{0}^{t}\bar\sigma(Z_{\varepsilon}(s))d^{-}B^{H}(s),$ then for a given arbitrarily small number $$\delta_1>0$$, there exist $$\varepsilon_1\in(0,\varepsilon_0]$$, such that for any $$\varepsilon\in(0,\varepsilon_1]$$, $$t\in[0,T]$$, $$E(| X_{\varepsilon}(t)-Z_{\varepsilon}(t)|^2)\leq\delta_1$$.

##### MSC:
 34F05 Ordinary differential equations and systems with randomness 34C29 Averaging method for ordinary differential equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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