文档介绍：Universitat¨ Fachbereich
Stuttgart Mathematik
Stochastic Differential Equations Driven by Gaussian
Processes with Dependent Increments
Jurgen¨ Dippon, Daniel Schiemert
Preprint 2006/001
Universitat¨ Stuttgart
Fachbereich Mathematik
Stochastic Differential Equations Driven by Gaussian
Processes with Dependent Increments
Jurgen¨ Dippon, Daniel Schiemert
Preprint 2006/001
Fachbereich Mathematik
Fakultat¨ Mathematik und Physik
Universitat¨ Stuttgart
Pfaffenwaldring 57
D-70 569 Stuttgart
E-Mail: ******@-
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ISSN 1613-8309
c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors.
LATEX-Style: Winfried Geis, Thomas Merkle
Stochastic Differential Equations Driven by Gaussian
Processes with Dependent Increments∗
Jurgen¨ Dippon and Daniel Schiemert
Stuttgart, Germany
February 10, 2006; corrected March 15, 2006
Abstract
We define an integral with respect to a class of centered Gaussian processes with
dependent increments. Furthermore, we consider stochastic differential equations
driven by such a process and discuss several examples. In the special case of a
bilinear stochastic differential equation existence and uniqueness of the solution is
proved. We derive a generalized Ornstein-Uhlenbeck process from an associated
stochastic differential equation. Finally, several applications are presented.
1 Introduction
H
In the last ten years fractional Brownian motion Bt gained a lot of attention (. [Be],
[GrNo], [HuOk], [HuOkSa], [So]). Opposed to Brownian motion this Gaussian process has
dependent increments. This is one of the reasons why it is interesting for applications
such as in finance (. [HuOk], [Be]) work simulations (. [No]). A disadvantage
H H
of fractional Brownian motion is that the shape of its covariance function E(Bt Bs ) de-
pends on a single parameter, the Hurst parameter H, only. For example, this restricted
flexibility in choosing