Stochastic averaging principle for dynamical systems with fractional Brownian motion.

*(English)*Zbl 1314.60122Summary: Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter \(H\) in the interval \((\frac{1}{2},1)\), is investigated. An averaged SDE for the original SDE is proposed, and their solutions are quantitatively compared. It is shown that the solution of the averaged SDE converges to that of the original SDE in the sense of mean square and also in probability. It is further demonstrated that a similar averaging principle holds for SDEs under stochastic integrals of pathwise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60G22 | Fractional processes, including fractional Brownian motion |

60H05 | Stochastic integrals |

34F05 | Ordinary differential equations and systems with randomness |

37H10 | Generation, random and stochastic difference and differential equations |

93E03 | Stochastic systems in control theory (general) |

##### Keywords:

stochastic differential equations; averaging principle; fractional Brownian motion; correlated noise; stochastic integrals##### References:

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