Trivariate Density of Brownian Motion, Its Local and Occupation Times, with Application to Stochastic Control

Abstract

We compute the joint density of Brownian motion, its local time at the origin, and its occupation time of $\lbrack 0, \infty)$. Two derivations of the main result are offered; one is computational, whereas the other uses some of the deep properties of Brownian local time. We use the result to compute the transition probabilities of the optimal process in a stochastic control problem.