Perfect cocycles through stochastic differential equations
Suppose \(\varphi\) is a continuous random dynamical system on \(\mathbb{R}^ d\). This is essentially a skew product flow consisting of random diffeomorphisms over \(\mathbb{R}^ d\), continuous in time and space, over a measurable base flow. The paper has two main results. First it is shown that \(\varphi\) is the solution of a stochastic differential equation with a vector field valued semimartingale with stationary increments as driving process, if and only if \(t \mapsto \varphi (t, \omega) x\) is a semimartingale for every \(x \in \mathbb{R}^ d\). This is worked out for two- sided time (where usually stochastic analysis is concerned with nonnegative time \(\mathbb{R}^ +\) only). Then it is proved that a (very general) crude cocycle with a locally compact group as `time', taking values in a countably generated Hausdorff group, can be perfected.
scientific article; zbMATH DE number 739104
scientific article