An inverse problem for the first-passage place of some diffusion processes with random starting point

Abstract

We study an inverse problem for the first-passage place of a one-dimensional diffusion process X(t) (also with jumps), starting from a random position $\eta \in [a,b].$ Let be ${\tau }_{a,b}$ the first time at which X(t) 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 η (if it exists) such that ${\pi }_a=q.$ Some explicit examples are reported.

Keywords:

• Jump-diffusion process
• first-passage place
• inverse first-passage place problem

Mathematics Subject Classification:

• 60J60
• 60H05
• 60H10

Acknowledgments

The author would like to express particular thanks to the anonymous reviewer for his/her useful comments, leading to improved presentation.

Additional information

Funding

He acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.