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Stochastic differential equations with fractal noise. (English) Zbl 1075.60075
A stochastic differential equation in $\Bbb{R}^n$, $$dX(t)=\sum_{j=0}^m a_j(X(t),t)dZ^j(t)+b(X(t),t)\,dt,\quad X(t_0)=X_0,$$ is considered where $Z^0$ is a continuous process with generalised bracket and $Z^1,\dots, Z^m$ are processes with sample paths in a fractional Sobolev space $W_2^\beta$ for some $\beta>1/2$. Stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed. It combines the stochastic Itô calculus with fractional norm estimates of associated integral operators in $W^\alpha_2$ for $\alpha\in (0,1)$. Linear equations are considered as a special case. This approach leads to fast computer algorithms based on Picard’s iteration method. See the author [Math. Nachr. 225, 145--183 (2001; Zbl 0983.60054)] for the case $m=0$ and {\it L. Coutin} and {\it L. Decreusefond} [Ann. Appl. Probab. 9, No. 4, 1058--1090 (1999; Zbl 0956.60058)] for the case of Gaussian driving processes.

60H10Stochastic ordinary differential equations
60H05Stochastic integrals
60H15Stochastic partial differential equations
34F05ODE with randomness
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