First Passage Times of a Jump Di usion Process

This paper studies the rst passage times to at boundaries for a double exponential jump di usion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the rst passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump di usion processes, the double exponential jump di usion process o ers a rare case in which analytical solutions for the rst passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The nance applications include pricing barrier and lookback options.