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

TL;DR

This paper presents additional explicit solutions to an inverse first-passage problem for one-dimensional jump-diffusion processes, focusing on finding initial distributions that yield a specified exit probability from an interval.

Contribution

It extends previous work by providing new explicit solutions to the inverse first-passage problem for jump-diffusion processes.

Findings

Derived explicit solutions for the inverse problem.

Identified conditions for the existence of initial distributions.

Enhanced understanding of exit probabilities in jump-diffusions.

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.