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Preprints, Working Papers, ... Year : 2011

Lévy process conditioned by its height process

Abstract

In the present work, we consider spectrally positive Lévy processes $(X_t,t\geq0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $\pfl$ of this conditioned process is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is $\left(\int\tilde\rt(\mathrm{d}z)e^{\alpha z}+I_t\right)\2{t\leq T_0}$ for some $\alpha$ and where $(I_t,t\geq0)$ is the past infimum process of $X$, where $(\tilde\rt,t\geq0)$ is the so-called \emph{exploration process} defined in \cite{Duquesne2002} and where $T_0$ is the hitting time of 0 for $X$. Under $\pfl$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of $\pfl$ as $x\to0$. When the process $X$ is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of $X$. The computations are easier in this case because $X$ can be viewed as the contour process of a (sub)critical \emph{splitting tree}. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.
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Dates and versions

hal-00599921 , version 1 (11-06-2011)
hal-00599921 , version 2 (30-01-2012)

Identifiers

  • HAL Id : hal-00599921 , version 2

Cite

Mathieu Richard. Lévy process conditioned by its height process. 2011. ⟨hal-00599921v2⟩
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