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Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE’s. (English) Zbl 1099.60040

Summary: We consider a stochastic process \(X_t^x\) which solves an equation \[ dX_t^x = AX_t^x\,dt + \Phi \,dB^H_t,\quad X_0^x=x, \] where \(A\) and \(\Phi \) are real matrices and \(B^H\) is a fractional Brownian motion with Hurst parameter \(H \in (1/2,1)\). The Kolmogorov backward equation for the function \(u(t,x) = \mathbb E f(X^x_t)\) is derived and exponential convergence of probability distributions of solutions to the limit measure is established.

MSC:

60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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References:

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