The first passage time of a stable process conditioned to not overshoot

TL;DR

This paper provides a new elementary proof and a novel expression for the density of the first passage time of a stable Lévy process, linking it to the process conditioned to not overshoot and to dual processes.

Contribution

It introduces an alternative proof of the absolute continuity of the first passage time law and derives a new density expression using the limit process as overshoot tends to zero.

Findings

New expression for the density of the first passage time

Relation between the limit passage time and the dual process

Elementary approach providing new insights

Abstract

Consider a stable L\'evy process $X=(X_t,t\geq 0)$ and let $T_x$, for $x>0$, denote the first passage time of $X$ above the level $x$. In this work, we give an alternative proof of the absolute continuity of the law of $T_x$ and we obtain a new expression for its density function. Our approach is elementary and provides a new insight into the study of the law of $T_x$. The random variable $T_x^0$, defined as the limit of $T_x$ when the corresponding overshoot tends to $0$, plays an important role in obtaining these results. Moreover, we establish a relation between the random variable $T_x^0$ and the dual process conditioned to die at $0$. This relation allows us to link the expression of the density function of the law of $T_x$ presented in this paper to the already known results on this topic.