On an Inverse First-Passage Problem for Jump-Diffusion Processes

Computing the exact mathematical expression for a quantity defined in terms of a first-passage time random variable for a jump-diffusion process is in general very difficult. In this paper, we consider the following inverse problem: can we find a certain distribution for the size of the jumps that leads to a simple solution of the integro-differential equation satisfied by the quantity of interest, subject to the appropriate boundary conditions? Such distributions are found, in particular, for the mean of the first-passage time for important jump-diffusion processes.