Perfect cocycles through stochastic differential equations

Abstract

We prove that if φ{symbol} is a random dynamical system (cocycle) for which t→φ{symbol}(t, ω)x is a semimartingale, then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment (helix), and conversely. This relation is succinctly expressed as "semimartingale cocycle=exp(semimartingale helix)". To implement it we lift stochastic calculus from the traditional one-sided time ℝ to two-sided time T=ℝ and make this consistent with ergodic theory. We also prove a general theorem on the perfection of a crude cocycle, thus solving a problem which was open for more than ten years. © 1995 Springer-Verlag.