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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Ordinary differential equations
with fractal noise


Authors: F. Klingenhöfer and M. Zähle
Journal: Proc. Amer. Math. Soc. 127 (1999), 1021-1028
MSC (1991): Primary 34A05; Secondary 60H10, 26A42
MathSciNet review: 1486738
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Abstract: The differential equation

\begin{displaymath}dx(t) \, = \, a(x(t),t) \,dZ(t) \:+\: b(x(t),t) \,dt \end{displaymath}

for fractal-type functions \begin{math}Z(t) \end{math} is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation \begin{math}x(t)\, =\, h(y(t)+Z(t),t) \end{math} for certain \begin{math}C^1 \end{math}-functions \begin{math}h \end{math} and \begin{math}y \end{math}. The method is also applied to Itô stochastic differential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions.


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Additional Information

F. Klingenhöfer
Affiliation: Mathematical Institute, University of Jena, D-07740 Jena, Germany
Email: klingenhofer@minet.uni-jena.de

M. Zähle
Affiliation: Mathematical Institute, University of Jena, D-07740 Jena, Germany
Email: zaehle@minet.uni-jena.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04803-0
PII: S 0002-9939(99)04803-0
Received by editor(s): July 9, 1997
Dedicated: To the memory of Johannes Kerstan
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society