Stochastic differential equations with fractal noise. (English) Zbl 1075.60075

A stochastic differential equation in \(\mathbb{R}^n\), \[ dX(t)=\sum_{j=0}^m a_j(X(t),t)dZ^j(t)+b(X(t),t)\,dt,\quad X(t_0)=X_0, \] 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 L. Coutin and L. Decreusefond [Ann. Appl. Probab. 9, No. 4, 1058–1090 (1999; Zbl 0956.60058)] for the case of Gaussian driving processes.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
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