Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control (Q796903)

Using the Feynman-Kac formula and Laplace transform the authors compute the joint density of Brownian motion, its local time at the origin, and its occupation time of \([0,\infty)\), with zero and nonzero initial condition. From these results and the Girsanov transformation the transition probabilities of a Brownian motion whose drift switches between two values as the process crosses a threshold is obtained.