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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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