A Wick functional limit theorem and applications to fractional Brownian motion

Parczewski, Peter

DOI: https://doi.org/10.22028/D291-26496
URL: https://publikationen.sulb.uni-saarland.de/handle/...
Document Type: Doctoral dissertation
Year of publication: 2013
Place of publication: Saarbrücken
University: Universität des Saarlandes
Evaluator: Bender, Christian
Date of oral examination: 2013
Publication language: English
Institution: School of Business Informatics and Mathematics > Wirtschaftsmathematik II (Neuenkirch 2013-)
Subject: 510 Mathematics
Keywords (English): stochastic analysis , fractional Brownian motion , Wick calculus , Wiener chaos decomposition
Abstract: 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 decompositions

Dieser Datensatz wurde nicht während einer Tätigkeit an der Universität Mannheim veröffentlicht, dies ist eine Externe Publikation.

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