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Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's

Michal Vyoral — 2005

Applications of Mathematics

We consider a stochastic process $X_{t}^{x}$ which solves an equation $d X_{t}^{x} = A X_{t}^{x} d t + Φ d B_{t}^{H}, X_{0}^{x} = x$ where $A$ and $Φ$ 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 $u (t, x) = 𝔼 f (X_{t}^{x})$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established.

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