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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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