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Ordinary differential equations with fractal noise

Author(s): F. Klingenhöfer; M. Zähle
Journal: Proc. Amer. Math. Soc. 127 (1999), 1021-1028.
MSC (1991): Primary 34A05; Secondary 60H10, 26A42
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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.

References:

1.: I. Karatzas and E. Shreve, Brownian motion and stochastic calculus. Springer, New York, 1991. MR 92h:60127
2.: T.G. Lyons, Differential equations driven by rough signals (I): an extension of an inequality by L.C. Young, Mathematical Research Letters 1 (1994), 451-464. MR 96b:60150
3.: S.G. Samko, A.A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives.Theory and Applications. Gordon and Breach, New York, 1993. MR 96d:26012
4.: M. Zähle, Integration with Respect to Fractal Functions and Stochastic Calculus, Probab. Theory Related Fields (to appear).


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