A Wick functional limit theorem and applications to fractional Brownian motion

The Wick product is a well-known tool in stochastic analysis to construct stochastic integrals with respect to Gaussian processes beyond semimartingales. Similarly, on disturbed random walks one can define a discrete counterpart. In this thesis we prove that weak convergence of central limit theorems carries over to applications of Wick products. Thus, the analogy of the discrete and continuous Wick calculus finds its expression in particular in convergence results. These convergences range to a functional limit theorem for Gaussian processes. Due to an extension of Sottinen's Donsker-type approximation of the fractional Brownian motion (Finance and Stochastics. (5), 343-355 (2001)) to all Hurst parameters, we can also approximate processes of fractional geometric Brownian motion type. Based on this, we examine the convergence of solutions of Wick difference equations to solutions of corresponding Wick-Ito stochastic differential equations. We determine the asymptotical computational costs of the difference equations and illustrate it on examples for the fractional Black-Scholes model. Moreover, we provide the equivalence conditions for convergence of discrete S-transforms to continuous S-transforms. In particular, this convergence is represented in terms of the Wiener chaos decomposition