On exit time of stable processes

TL;DR

This paper analyzes the exit time of one-dimensional stable processes, deriving explicit formulas for its Laplace transform and density, and revealing convolution relations, with specific results for symmetric 2/3-stable processes.

Contribution

It provides explicit formulas for the Laplace transform and density of exit times for stable processes, including special cases like the symmetric 2/3-stable process.

Findings

Explicit Laplace transform expressions for exit times.

Derived density functions for specific stable processes.

Identified multiplicative convolution relations for exit times.

Abstract

We study the exit time $\tau=\tau_{(0,\infty)}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^\alpha$ as the Laplace transform of a positive random variable with explicit density. Consequently, $\tau$ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric $\frac23$-stable process, explicit formulas for the Laplace transform and the density of $\tau$ are obtained as an application.