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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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References:
 [1] Alòs, E.; Nualart, D., Stochastic integration with respect to the fractional Brownian motion, Stochast. Stochast. Rep., 75, 129-152, (2003) · Zbl 1028.60048 [2] Biagini, F.; Hu, Y.; Øksendal, B.; Zhang, T., Stochastic Calculus for Fractional Brownian Motion and Applications, (2008), Springer-Verlag [3] Chakravarti, N.; Sebastian, K. L., Fractional Brownian motion models for polymers, Chem Phys Lett., 267, 9-13, (1997) [4] Cox, J. C.; Ingersoll, J. E.; Ross, S. A., A theory of the term structure of interest rates, Econometrica., 53, 385-407, (1985) · Zbl 1274.91447 [5] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion I. theory, SIAM. J. Control. Optim., 38, 582-612, (2000) · Zbl 0947.60061 [6] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Fin. Stud., 6, 327-343, (1993) · Zbl 1384.35131 [7] Hu, Y.; Peng, S., Backward stochastic differential equation driven by fractional Brownian motion, SIAM. J. Control. Optim., 48, 1675-1700, (2009) · Zbl 1284.60117 [8] Hurst, H. E., Long-term storage capacity in reservoirs, Trans. Amer. Soc. Civil. Eng., 116, 400-410, (1951) [9] Jaczak-Borkowska, K., Generalized BSDEs driven by fractional Brownian motion, Stat. Probab. Lett., 83, 805-811, (2013) · Zbl 1267.60062 [10] Khasminskii, R. Z., A limit theorem for the solution of differential equations with random right-hand sides, Theory. Probab. Appl., 11, 390-405, (1963) [11] Kolmogorov, A. N., Wienersche spiralen und einige andere interessante kurven im hilbertschen, Raum. C. R. Acad. Sci. URSS., 26, 115-118, (1940) · JFM 66.0552.03 [12] Kolomiets, V. G.; Mel’nikov, A. I., Averaging of stochastic systems of integral-differential equations with Poisson noise, Ukr. Math. J., 2, 242-246, (1991) · Zbl 0735.60060 [13] Kwok, Y. K., Pricing multi-asset options with an external barrier, Int. J. Theor. Appl. Fin., 1, 523-541, (1998) · Zbl 0987.91030 [14] Lan, G.; Wu, J. L., New sufficient conditions of existence, moment estimations and non confluence for SDEs with non-Lipschitzian coefficients, Stoch. Proc. Appl., 124, 4030-4049, (2014) · Zbl 1314.60116 [15] Luo, J.; Taniguchi, T., The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps, Stoch. Dyn., 9, 135-152, (2009) · Zbl 1167.60336 [16] Mandelbrot, B. B.; Van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-427, (1968) · Zbl 0179.47801 [17] Pei, B.; Xu, Y., Mild solutions of local non-Lipschitz stochastic evolution equations with jumps, Appl. Math. Lett., 52, 80-86, (2016) · Zbl 1356.60106 [18] Russo, F.; Vallois, P., Forward, backward and symmetric stochastic integration, Probab. Theory Relat. Field, 97, 403-421, (1993) · Zbl 0792.60046 [19] Scheffer, R.; Maciel, F. R., The fractional Brownian motion as a model for an industrial airlift reactor, Chem. Eng. Sci., 56, 707-711, (2001) [20] Stoyanov, I. M.; Bainov, D. D., The averaging method for a class of stochastic differential equations, Ukr. Math. J., 26, 186-194, (1974) · Zbl 0294.60051 [21] Taniguchi, T., Successive approximations to solutions of stochastic differential equations, J. Differential Equations, 96, 152-169, (1992) · Zbl 0744.34052 [22] Taniguchi, T., The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps, Stoch., 82, 339-363, (2010) · Zbl 1219.60064 [23] Taniguchi, T., The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations, J. Math. Anal. Appl., 340, 197-208, (2009) [24] Xu, Y.; Duan, J.; Xu, W., An averaging for stochastic dynamical systems with Lévy noise, Physica D., 240, 1395-1401, (2011) · Zbl 1236.60060 [25] Xu, Y.; Pei, B.; Li, Y. G., Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci., 38, 2120-2131, (2015) · Zbl 1345.60051 [26] Xu, Y., Stochastic averaging for dynamical systems with fractional Brownian motion, Discr. Cont. Dyn-B, 19, 1197-1212, (2014) · Zbl 1314.60122 [27] Xu, Y.; Pei, B.; Guo, R., Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discr. Cont. Dyn-B, 20, 2257-2267, (2015) · Zbl 1335.34090 [28] Xu, Y.; Pei, B.; Li, Y. G., An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstr. Appl. Anal., 2014, 479195, (2014) [29] Yamada, T., On the successive approximation of solutions of stochastic differential equations, J. Math. Kyoto Univ., 21, 501-515, (1981) · Zbl 0484.60053
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