## 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.

### MSC:

 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

### Citations:

Zbl 0983.60054; Zbl 0956.60058
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### References:

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