Lévy processes conditioned on having a large height process

Mathieu Richard

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 982-1013
  • ISSN: 0246-0203

Abstract

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In the present work, we consider spectrally positive Lévy processes not drifting to and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with ) before hitting . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law of this conditioned process (starting at ) is defined as a Doob -transform via a martingale. For Lévy processes with infinite variation paths, this martingale is for some and where is the past infimum process of , where is the so-calledexploration processdefined in [10] and where is the hitting time of 0 for . Under , we also obtain a path decomposition of at its minimum, which enables us to prove the convergence of as . When the process is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of . The computations are easier in this case because can be viewed as the contour process of a (sub)criticalsplitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.

How to cite

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Richard, Mathieu. "Lévy processes conditioned on having a large height process." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 982-1013. <http://eudml.org/doc/271942>.

@article{Richard2013,
abstract = {In the present work, we consider spectrally positive Lévy processes $(X_\{t\},t\ge 0)$ 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 $\{\mathbb \{P\}\}_\{x\}^\{\star \}$ of this conditioned process (starting at $x&gt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is $(\int \tilde\{\rho \}_\{t\}(\mathrm \{d\}z)\mathrm \{e\}^\{\{\alpha \}\{z\}\}+I_\{t\})\mathbf \{1\} _\{\lbrace t\le T_\{0\}\rbrace \}$ for some $\alpha $ and where $(I_\{t\},t\ge 0)$ is the past infimum process of $X$, where $(\tilde\{\rho \}_\{t\},t\ge 0)$ is the so-calledexploration processdefined in [10] and where $T_\{0\}$ is the hitting time of 0 for $X$. Under $\{\mathbb \{P\}\}_\{x\}^\{\star \}$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of $\{\mathbb \{P\}\}_\{x\}^\{\star \}$ as $x\rightarrow 0$. 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)criticalsplitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.},
author = {Richard, Mathieu},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Lévy process; height process; Doob harmonic transform; splitting tree; spine decomposition; Size-biased distribution; queueing theory; size-biased distribution; exploration process},
language = {eng},
number = {4},
pages = {982-1013},
publisher = {Gauthier-Villars},
title = {Lévy processes conditioned on having a large height process},
url = {http://eudml.org/doc/271942},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Richard, Mathieu
TI - Lévy processes conditioned on having a large height process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 982
EP - 1013
AB - In the present work, we consider spectrally positive Lévy processes $(X_{t},t\ge 0)$ 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 ${\mathbb {P}}_{x}^{\star }$ of this conditioned process (starting at $x&gt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is $(\int \tilde{\rho }_{t}(\mathrm {d}z)\mathrm {e}^{{\alpha }{z}}+I_{t})\mathbf {1} _{\lbrace t\le T_{0}\rbrace }$ for some $\alpha $ and where $(I_{t},t\ge 0)$ is the past infimum process of $X$, where $(\tilde{\rho }_{t},t\ge 0)$ is the so-calledexploration processdefined in [10] and where $T_{0}$ is the hitting time of 0 for $X$. Under ${\mathbb {P}}_{x}^{\star }$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of ${\mathbb {P}}_{x}^{\star }$ as $x\rightarrow 0$. 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)criticalsplitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.
LA - eng
KW - Lévy process; height process; Doob harmonic transform; splitting tree; spine decomposition; Size-biased distribution; queueing theory; size-biased distribution; exploration process
UR - http://eudml.org/doc/271942
ER -

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