arXiv subjects

Juan Carlos Pardo Millan

Publications and source records attributed to Juan Carlos Pardo Millan.

4 recordsLinked to original sources

arXiv2013

On the extinction of Continuous State Branching Processes with catastrophes

We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Lévy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of Lévy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.

math.PR↗

arXiv2007

Some explicit identities associated with positive self-similar Markov processes

We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type $π(dx)=e^{γx}ν(e^x-1) dx$, where $ν$ is the density of the stable Lévy measure and $γ$ is a positive parameter which depends on its characteristics. These processes were introduced in \cite{CC} as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. In this paper, we compute explicitly the law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points.

math.PR↗

arXiv2007

On the genealogy on conditioned stable Lévy forest

We give a realization of the stable Lévy forest of a given size conditioned by its mass from the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering $k$ independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to $n$. We prove that when $n$ and $k$ tend towards $+\infty$, under suitable rescaling, the associated coding random walk, the contour and height processes converge in law on the Skorokhod space respectively towards the "first passage bridge" of a stable Lévy process with no negative jumps and its height process.

math.PR↗

arXiv2007

The upper envelope of positive self-similar Markov processes

We establish integral tests and laws of the iterated logarithm at 0 and at $+\infty$, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation, time reversal arguments and on the study of the upper envelope of their future infimum due to Pardo \cite{Pa}. These results extend integral test and laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdös \cite{de} and stable Lévy processes conditioned to stay positive with no positive jumps due to Bertoin \cite{be1}.

math.PR↗