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Zbl 0758.60073
Bertoin, Jean
Sur la décomposition de la trajectoire d'un processus de Lévy spectralement positif en son infimum. (On the path decomposition at the infimum for a spectrally positive Lévy process). (French)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 27, No.4, 537-547 (1991). ISSN 0246-0203

Let $X$ be the coordinate process on the space $\Omega$ of càdlàg functions with a life time $\zeta$ and the natural filtration $({\cal F}\sb t$, $t\geq 0)$, $\overline {X}\sb t=\sup\{X\sb s$, $0\leq s\leq t\}$, $\underline {X}\sb t=\inf\{X\sb s$, $0\leq s\leq t\}$, $\underline{\underline {X}}\sb t=\inf\{X\sb s$, $t\leq s<\zeta\}$, $\rho=\sup\{t\geq 0$: $\underline{\underline {X}}\sb t=\underline {X}\sb \infty\}$, $\tau(-x)=\inf\{t\geq 0$: $X\sb t<-x\}$. It is considered a probability measure $P$ on ${\cal F}\sb \infty$ such that $X$ is a spectrally positive Lévy process starting from 0 and drifting to $+\infty$. Denoting $\alpha$ the unique positive solution of $\psi(\alpha)=0$, where $\psi$ is the cumulant function of $X$, define the probability measure $P\sp*$ by means of the equality $$dP\sp*\mid\sb{{\cal F}\sb t}=\exp\{-\alpha X\sb t\}dP\mid\sb{{\cal F}\sb t}, \qquad t\geq 0.$$ It is proved that the probability distribution of the preinfimum process $(X\sb t$, $0\leq t<\rho)$ with respect to $P$ coincides with the probability distribution of the process $(X\sb t$, $0\leq t<\tau(-\gamma))$ with respect to $P\sp*$, where $\gamma$ is independent of $X$ and exponentially distributed with the parameter $\alpha$ and the probability distribution of $X- 2\underline{\underline {X}}\sp c-{\cal J}$ with respect to $P$ equals to $P\sp*$, where $\underline{\underline {X}}\sp c$ denotes the continuous part of $\underline{\underline {X}}$ and $${\cal J}\sb t=\sum\sb{0<s\leq t}(X\sb s-X\sb{s-})\text{{\bf 1}}\sb{\{\underline{\underline {X}}\sb s>X\sb{s-}\}}.$$ The known inverse Pitman's theorem is extended.
[B.Grigelionis (Vilnius)]

MSC 2000:: *60J99 Markov processes
60G30 Induced measures of stochastic processes

Keywords: path decomposition; spectrally positive Lévy process; cumulant function; inverse Pitman's theorem

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