The first-crossing area of a diffusion process with jumps over a constant barrier

TL;DR

This paper investigates the distribution and moments of the integral of a jump-diffusion process until its first crossing of a barrier, extending previous work on diffusion areas and providing explicit examples.

Contribution

It introduces PDE-difference equations for the Laplace transform and moments of the area under jump-diffusions until first crossing, extending prior diffusion results.

Findings

Laplace transform and moments satisfy PDE-difference equations.

Distribution of the process's minimum is characterized.

Explicit examples for diffusions with and without jumps are provided.

Abstract

For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$ starting from $x<S,$ we study the probability distribution of the integral $A_S(x)= \int_0 ^ {\tau_S(x)}X(t) \ dt$ determined by $X(t)$ till its first-crossing time $\tau_S(x)$ over $S.$ In particular, we show that the Laplace transform and the moments of $A_S(x)$ are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of $X(t)$ in $[0, \tau_S(x)]$ is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by $X(t)$ till its first-passage below zero. Some explicit examples are reported, regarding diffusions with and without jumps.