Peter Parczewski

Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus

We approximate the solution of some linear systems of SDEs driven by a fractional Brownian motion

A Fractional Donsker Theorem

B^H

Donsker-type theorems for correlated geometric fractional Brownian motions and related processes

with Hurst parameter

The self-normalized Donsker theorem revisited

H\in(\frac{1}{2},1)

A Wick functional limit theorem and applications to fractional Brownian motion

in the Wick--It\^{o} 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 the 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

On the connection between discrete and continuous Wick calculus with an application to the fractional Black-Scholes model

B^H

Optimal Approximation of Skorohod Integrals

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Discretizing Malliavin calculus

We prove a Donsker-type approximation of the fractional Brownian motion which extends a result by Sottinen for the case H > 1/2 to the full range of Hurst parameters H ∈ (0, 1). The convergence is established by a Donsker-type theorem for Volterra Gaussian processes. The approximation is applied to weak convergence of fractional Wiener integrals.

Extensions of the Hitsuda–Skorokhod Integral

We prove a Donsker-type theorem for vector processes of functionals of correlated Wiener integrals. This includes the case of correlated geometric fractional Brownian motions of arbitrary Hurst parameters in (0,1) driven by the same Brownian motion. Starting from a Donsker-type approximation of Wiener integrals of Volterra type by disturbed binary random walks, the continuous and discrete Wiener chaos representation in terms of Wick calculus is effective. The main result is the compatibility of these continuous and discrete stochastic calculi via these multivariate limit theorems.

Optimal Approximation of Skorohod Integrals – Examples with Substandard Rates

We extend the Poincare–Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space