SMILE

Stochastic Models for the Inference of Life Evolution

Lévy processes conditioned on having a large height process

Richard, M.

Annales de l'Institut Henri Poincaré

2013

Bibtex

@article{richard_levy_2013,
Author = {Richard, Mathieu},
Title = {Lévy processes conditioned on having a large height
process},
Journal = {Annales de l'Institut Henri Poincaré},
Volume = {49},
Number = {4},
Pages = {982--1013},
abstract = "In the present work, we consider spectrally positive
L\'evy processes \$(X\_t,t\backslash{}geq0)\$ not
drifting to \$+\backslash{}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\backslash{}'evy
processes to stay positive. The (honest) law
\$\backslash{}pfl\$ of this conditioned process is
defined as a Doob \$h\$-transform via a martingale. For
L\backslash{}'evy processes with infinite variation
paths, this martingale is
\$(\backslash{}int\backslash{}tilde\backslash{}rt(\backslash{}mathrm\{d\}z)e^\{\backslash{}alpha
z\}+I\_t)\backslash{}2\{t\backslash{}leq T\_0\}\$ for
some \$\backslash{}alpha\$ and where
\$(I\_t,t\backslash{}geq0)\$ is the past infimum
process of \$X\$, where
\$(\backslash{}tilde\backslash{}rt,t\backslash{}geq0)\$
is the so-called \backslash{}emph\{exploration
process\} defined in Duquesne, 2002, and where \$T\_0\$
is the hitting time of 0 for \$X\$. Under
\$\backslash{}pfl\$, we also obtain a path
decomposition of \$X\$ at its minimum, which enables us
to prove the convergence of \$\backslash{}pfl\$ as
\$x\backslash{}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 \backslash{}emph\{splitting
tree\}. We also can give an alternative
characterization of our conditioned process in the vein
of spine decompositions.",
doi = {10.1214/12-AIHP491},
year = 2013
}

Link to the article

Accéder à l'article grâce à son DOI.