On the First-Passage Area of a One-Dimensional Jump-Diffusion Process

Abstract

For a one-dimensional jump-diffusion process X(t), starting from x > 0, it is studied the probability distribution of the area A(x) swept out by X(t) till its first-passage time below zero. In particular, it is shown that the Laplace transform and the moments of A(x) are solutions to certain partial differential-difference equations with outer conditions. The distribution of the maximum displacement of X(t) is also studied. Finally, some explicit examples are reported, regarding diffusions with and without ju

Investigating the Distribution of the First-Crossing Area of a Diffusion Process with Jumps Over a Threshold

“…and it extends the results of a previous paper by the author ( [1]), concerning the area swept out by ) (t X till its first-passage below zero. As for results about the integral of ) (t X over a deterministic and fixed time interval, see e.g [2].…”

“…Let In the present article, we complete the study carried out in [1]; in fact, for a one-dimensional jump-diffusion process, in place of the first-passage area below zero, we consider the analogous problem of first-crossing area over a positive barrier.…”

“…S We improperly call Since the topic was studied quite extensively in [1] (though in the slight different situation of first passage below zero), we will omit some details. We will suppose that ) (t X is the solution of a stochastic differential equation of the form:…”

“…The general solution of (30) involves arbitrary constants 1 c and 2 c and, as easily seen, it is given by…”

One-Dimensional Reflected Diffusions with Two Boundaries and an Inverse First-Hitting Problem

“…without reflecting (see e.g. [6], [7], [8], [9], [10], [11], [15], [16], [19], [35], [40], [41]), more recently (see e.g. [14], [30], [38] ) some results appeared about the FPT of a one-dimensional reflected diffusion, through a threshold S.…”