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Differential equations driven by rough signals. (English) Zbl 0923.34056
The author deals with the equation \[ dy_t= \sum_i f^i(y_t) dx^i_t, \] where \(f^i\) are vectors fields, \(x_t\) represents some forcing or controlling term and the trajectory \(y_t\) represents some filtered effect thereof. Such equations are very common, in particular they appear in probability where the driving signal might be a vector-valued Brown motion, semi-martingale or similar process. Here, a systematic approach to such equation is given in the case when the driving signal \(x_t\) is a rough path. This approach consists in using of deterministic meaning of (stochastic) integration of rough paths.
The results are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. Variable step size algorithms for the numerical integration of stochastic equations are constructed as a consequence of these results.

MSC:
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93E03 Stochastic systems in control theory (general)
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