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

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.

To appear in Theory of Probability and Its Applications, 16 pages, 15 figures