Regularity of the Local Time of Diffusions on the Positive Real Line with Reflection at Zero

Abstract

We study the joint law of (Xt(x), Lt(x)) where Xt(x) is the solution of a one dimensional stochastic differential equation on (0,+∞) with reflection at zero, and Lt(x) is its local time. In particular, we give some representation formula of the distribution of (Xt(x), Lt(x)), and we investigate the regularity of the joint density with respect to the local time argument under ellipticity and mild regularity conditions on coefficients of Xt(x).