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- W4301888969 abstract "In the present work, we consider spectrally positive L'evy processes $(X_t,tgeq0)$ 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 L'evy processes to stay positive. The (honest) law $pfl$ of this conditioned process is defined as a Doob $h$-transform via a martingale. For L'evy processes with infinite variation paths, this martingale is $(inttildert(mathrm{d}z)e^{alpha z}+I_t)2{tleq T_0}$ for some $alpha$ and where $(I_t,tgeq0)$ is the past infimum process of $X$, where $(tildert,tgeq0)$ is the so-called emph{exploration process} defined in Duquesne, 2002, 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 $xto0$. 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." @default.
- W4301888969 created "2022-10-06" @default.
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- W4301888969 date "2011-06-11" @default.
- W4301888969 modified "2023-10-18" @default.
- W4301888969 title "L'evy processes conditioned on having a large height process" @default.
- W4301888969 doi "https://doi.org/10.48550/arxiv.1106.2245" @default.
- W4301888969 hasPublicationYear "2011" @default.
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