Path transformations for local times of one-dimensional diffusions

Abstract

Let X be a regular one-dimensional transient diffusion and Ly be its local time at y. The stochastic differential equation (SDE) whose solution corresponds to the process X conditioned on [L∞y=a] for a given a≥0 is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction Bessel-type motions as well as their SDE representations are studied. Moreover, the Engelbert–Schmidt theory for the weak solutions of one dimensional SDEs is extended to the case when the initial condition is an entrance boundary for the diffusion. This extension was necessary for the construction of the Bessel-type motion which played an essential part in the SDE representation of X conditioned on [L∞y=a].