Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

Abstract

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If $X(t)$ is a one-dimensional diffusion with jumps, starting from a random position $\eta \in [a,b],$ let be $\tau_{a,b}$ the time at which $X(t)$ first exits the interval $(a,b),$ and $\pi _a = P(X(\tau_{a,b}) \le a)$ the probability of exit from the left of $(a,b).$ Given a probability $q \in (0,1),$ the problem consists in finding the density $g$ of $\eta$ (if it exists) such that $\pi _a = q;$ it can be seen as a problem of optimization.