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Stochastic averaging principle for dynamical systems with fractional Brownian motion. (English) Zbl 1314.60122
Summary: 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)
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