Ordinary differential equations with fractal noise
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- by F. Klingenhöfer and M. Zähle PDF
- Proc. Amer. Math. Soc. 127 (1999), 1021-1028 Request permission
Abstract:
The differential equation \[ dx(t) = a(x(t),t) dZ(t) \:+\: b(x(t),t) dt \] for fractal-type functions $Z(t)$ is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation $x(t) = h(y(t)+Z(t),t)$ for certain $C^1$-functions $h$ and $y$. 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.References
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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
- Received by editor(s): July 9, 1997
- Communicated by: Hal L. Smith
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1021-1028
- MSC (1991): Primary 34A05; Secondary 60H10, 26A42
- DOI: https://doi.org/10.1090/S0002-9939-99-04803-0
- MathSciNet review: 1486738
Dedicated: To the memory of Johannes Kerstan