Persistence and exit times for some additive functionals of skew Bessel processes

Let X be some homogeneous additive functional of a skew Bessel process Y. In this note, we compute the asymptotics of the first passage time of X to some fixed level b, and study the position of Y when X exits a bounded interval [a, b]. As a by-product, we obtain the probability that X reaches the level b before the level a. Our results extend some previous works on additive functionals of Brownian motion by Isozaki and Kotani for the persistence problem, and by Lachal for the exit time problem

Interfacial Phenomena and Natural Local Time

This article addresses a modification of local time for stochastic processes, to be referred to as `natural local time'. It is prompted by theoretical developments arising in mathematical treatments of recent experiments and observations of phenomena in the geophysical and biological sciences pertaining to dispersion in the presence of an interface of discontinuity in dispersion coefficients. The results illustrate new ways in which to use the theory of stochastic processes to infer macro scale parameters and behavior from micro scale observations in particular heterogeneous environments

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations

Continuity of Local Time: An applied perspective

Continuity of local time for Brownian motion ranks among the most notable mathematical results in the theory of stochastic processes. This article addresses its implications from the point of view of applications. In particular an extension of previous results on an explicit role of continuity of (natural) local time is obtained for applications to recent classes of problems in physics, biology and finance involving discontinuities in a dispersion coefficient. The main theorem and its corollary provide physical principles that relate macro scale continuity of deterministic quantities to micro scale continuity of the (stochastic) local time.Comment: To appear in: "The fascination of Probability, Statistics and Their Applications. In honour of Ole E. Barndorff-Nielsen on his 80th birthday

Passage-time statistics of superradiant light pulses from Bose-Einstein condensates

We discuss the passage-time statistics of superradiant light pulses generated during the scattering of laser light from an elongated atomic Bose-Einstein condensate. Focusing on the early-stage of the phenomenon, we analyze the corresponding probability distributions and their scaling behaviour with respect to the threshold photon number and the coupling strength. With respect to these parameters, we find quantities which only vary significantly during the transition between the Kapitza Dirac and the Bragg regimes. A possible connection of the present observations to Brownian motion is also discussed.Comment: Close to the version published in J. Phys.

Corrections for "Occupation and local times for skew Brownian motion with applications to dispersion across an interface"

We are making corrections and acknowledging colleagues that pointed out mistakes in our recent paper titled "Occupation and local times for skew Brownian motion with applications to dispersion across an interface" which was published in Annals of Applied Probability (2011) 21(1) 183-214. Specifically the corrections are: 1. The restriction of $\gamma$ to non negative values in Theorem 1.3 is not needed. But one has probabilistic interpretation only when $\gamma$ is non negative. 2. State the correct formulas in Corollary 3.3 as their were computational errors in the original formulas. We thank Pierre Etoir\'e and Miguel Martinez for pointing out these errors.Comment: Published in at http://dx.doi.org/10.1214/11-AAP775 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org