A note on the quasi-stationary distribution of the Shiryaev martingale on the positive half-line

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 [P. Collet, S. Martínez, and J. San Martín, Quasi-Stationary Distributions. Markov Chains, Diffusions and Dynamical Systems, Springer, Heidelberg, 2013] but has heretofore remained unsolved.