30 research outputs found
Parisian ruin probability for spectrally negative L\'{e}vy processes
In this note we give, for a spectrally negative Levy process, a compact
formula for the Parisian ruin probability, which is defined by the probability
that the process exhibits an excursion below zero, with a length that exceeds a
certain fixed period r. The formula involves only the scale function of the
spectrally negative Levy process and the distribution of the process at time r.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ404 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Smoothness of continuous state branching with immigration semigroups
In this work we develop an original and thorough analysis of the
(non)-smoothness properties of the semigroups, and their heat kernels,
associated to a large class of continuous state branching processes with
immigration. Our approach is based on an in-depth analysis of the regularity of
the absolutely continuous part of the invariant measure combined with a
substantial refinement of Ogura's spectral expansion of the transition kernels.
In particular, we find new representations for the eigenfunctions and
eigenmeasures that allow us to derive delicate uniform bounds that are useful
for establishing the uniform convergence of the spectral representation of the
semigroup acting on linear spaces that we identify. We detail several examples
which illustrate the variety of smoothness properties that CBI transition
kernels may enjoy and also reveal that our results are sharp. Finally, our
technique enables us to provide the (eventually) strong Feller property as well
as the rate of convergence to equilibrium in the total variation norm
Occupation times of intervals until first passage times for spectrally negative LĂ©vy processes
In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative LĂ©vy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented
Optimal control with absolutely continuous strategies for spectrally negative lévy processes
LĂ©vy processes