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Time reversal on Lévy processes. (English) Zbl 0646.60052
Let a complete probability space be given with two filtrations $$F=({\mathcal F}_ t)$$, $$\tilde H = (\tilde {\mathcal H}_ t)$$, $$t\in [0,1]$$. Let $$Y=(Y_ t)$$, $$t\in [0,1]$$, be a process with right continuous paths having left limits. We associate to Y the time-reversed process $$\tilde Y = (\tilde Y_ t)$$, $$t\in [0,1]$$, given by $\tilde Y_ t = \begin{cases} 0, & \text{if $$t=0$$} \\ Y_{(1-t)-}-Y_{1-}, & \text{if $$0<t<1$$} \\ Y_ 0- Y_{1-} ,& \text{if $$t=1$$} \end{cases}$ where $$Y_{u- }=\lim_{s\uparrow u}Y_ s$$, $$0<u\leq 1$$. Y is called an $$(F,\tilde H)$$- reversible semimartingale if (i) Y is an F-semimartingale on [0,1] and (ii) $$\tilde Y$$ is an $$\tilde H$$-semimartingale on [0,1).
Let Z be a Lévy process (a process with independent and stationary increments), Z c be its continuous martingale part. The authors prove that Z, $$Z^ c$$, $$\int^{t}_{0} f(Z_{s-})dZ_ s$$, $$\int^{t}_{0} f(Z_{s-})dZ^ c_ s$$ are reversible semimartingales for some functions f.
Reviewer: L.Gal’čuk

##### MSC:
 60G44 Martingales with continuous parameter 60H05 Stochastic integrals 60J99 Markov processes 60J65 Brownian motion
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