Diffusion occupation time before exiting

Abstract

Using the approach of D. Landriault et al. and B. Li and X. Zhou, for a one-dimensional time-homogeneous diffusion process X and constants c < a < b < d, we find expressions of double Laplace transforms of the form \(\mathbb{E}_x [e^{ - \theta T_d - \lambda } \int_0^{T_d } {1_{a < X_s < b} ds} ;T_d < T_c ]\), where T x denotes the first passage time of level x. As applications, we find explicit Laplace transforms of the corresponding occupation time and occupation density for the Brownian motion with two-valued drift and that of occupation time for the skew Ornstein-Uhlenbeck process, respectively. Some known results are also recovered.