A Note on the Quasi-Stationary Distribution of the Shiryaev Martingale on the Positive Half-Line

TL;DR

This paper derives a closed-form expression for the quasi-stationary distribution of the Shiryaev martingale diffusion on the positive half-line, solving a previously unsolved Sturm-Liouville problem.

Contribution

It provides an analytical solution for the quasi-stationary distribution of the Shiryaev martingale, addressing a longstanding open problem in stochastic process theory.

Findings

Closed-form formula for the quasi-stationary distribution

Solution of a singular Sturm-Liouville problem

Addresses an open problem from Collet et al. (2013)

Abstract

We obtain a closed-form formula for the quasi-stationary distribution of the classical Shiryaev martingale diffusion considered on the positive half-line $[A,+\infty)$ with $A>0$ fixed; the state space's left endpoint is assumed to be the killing boundary. The formula is obtained analytically as the solution of the appropriate singular Sturm-Liouville problem; the latter was first considered in Section 7.8.2 of Collet et al. (2013), but has heretofore remained unsolved.