Summary
We prove that if ϕ is a random dynamical system (cocycle) for whicht→ϕ(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 timeT=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.
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