Time reversal on Lévy processes
type: Item
instance of: scholarly article
title: Time reversal on Lévy processes
zbMATH Open document ID: 0646.60052
author: Jean Jacod
author: Philip Protter
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4054722
zbMATH Keywords: Lévy process
zbMATH Keywords: process with independent and stationary increments
zbMATH Keywords: reversible semimartingales
Wikidata QID: Q56907094
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991776
OpenAlex ID: W2062720516
DOI: 10.1214/AOP/1176991776
description: scientific article; zbMATH DE number 4054722