Law of two-sided exit by a spectrally positive strictly stable process

TL;DR

This paper derives explicit formulas for the distribution of exit times and related variables for a spectrally positive stable process, revealing independence properties and complex density structures.

Contribution

It provides new explicit density formulas for first exit times and joint distributions for spectrally positive stable processes with index in (1,2).

Findings

Density of first exit time expressed as an infinite sum involving roots of Mittag-Leffler functions.

Conditional independence of exit time and jump given the undershoot.

Explicit formulas for joint distributions of exit time, undershoot, and jump.

Abstract

For a spectrally positive strictly stable process with index in (1,2), the paper obtains i) the density of the time when the process makes first exit from an interval by hitting the interval's lower end point before jumping over its upper end point, and ii) the joint distribution of the time, the undershoot, and the jump of the process when it makes first exit the other way around. For i), the density of the time of first exit is expressed as an infinite sum of functions, each the product of a polynomial and an exponential function, with all coefficients determined by the roots of a Mittag-Leffler function. For ii), conditional on the undershoot, the time and the jump of first exit are independent, and the marginal conditional densities of the time has similar features as i).