The joint Laplace transforms for killed diffusion occupation times

The approach of Li and Zhou (2014) is adopted to find the Laplace transform of occupation time over interval (0, a) and joint occupation times over semi-infinite intervals (−∞, a) and (b, ∞) for a time-homogeneous diffusion process up to an independent exponential time e_q for 0 < a < b. The results are expressed in terms of solutions to the differential equations associated with the diffusion generator. Applying these results, we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift.