Probability Bifurcations of Lévy Bridges

Abstract

A Lévy bridge-a stable Lévy stochastic process conditioned to arrive at some state at some later time-can exhibit behavior differing dramatically from the more widely studied case of conditioned Brownian (Gaussian) processes. This difference stems from a structural change in the conditioned probability density at intermediate times as the arrival position varies. This structural shift gives rise to a distinction between "short" and "long" jumps. We explore the consequences of this idea for the statistics of Lévy vs. Brownian bridges, with applications to the analysis of the boundarycrossing problem and a computationally useful representation of Lévy bridges that does not carry over directly from the Gaussian case.