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Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus. (English) Zbl 1248.60044
Summary: We approximate the solution of some linear systems of SDEs driven by a fractional Brownian motion \(B^{H}\) with Hurst parameter \(H\in (\frac{1}{2} , 1)\) in the Wick-Itô sense, including a geometric fractional Brownian motion. To this end, we apply a Donsker-type approximation of the fractional Brownian motion by disturbed binary random walks due to Sottinen. Moreover, we replace the rather complicated Wick products by their discrete counterpart, acting on binary variables, in the corresponding systems of Wick difference equations. As the solutions of the SDEs admit series representations in terms of Wick powers, a key to the proof of our Euler scheme is an approximation of the Hermite recursion formula for the Wick powers of \(B^{H}\).

MSC:
60G22 Fractional processes, including fractional Brownian motion
60B10 Convergence of probability measures
60H07 Stochastic calculus of variations and the Malliavin calculus
60H40 White noise theory
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