A Note on the Quasi-Stationary Distribution of the Shiryaev Martingale on the Positive Half-Line
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We obtain a closed-form formula for the quasi-stationary distribution of the classical Shiryaev martingale diffusion considered on the positive half-line [A,+∞) 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.
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