On the quasi-stationary distribution of the Shiryaev–Roberts diffusion

We consider the diffusion generated by the equation with fixed, and where μ≠0 is given, and is standard Brownian motion. We assume that is stopped at with A>0 preset, and obtain a closed-from formula for the quasi-stationary distribution of , i.e., the limit , x∈[0,A]. Further, we also prove QA(x) to be unimodal for any A>0, and obtain its entire moment series. More importantly, the pair with r≥0 and A>0 is the well-known Generalized Shiryaev–Roberts change-point detection procedure, and its characteristics for r∼QA(x) are of particular interest, especially when A>0 is large. In view of this circumstance, we offer an order-three large-A asymptotic approximation of QA(x) valid for all x∈[0,A]. The approximation is rather accurate even if A is lower than what would be considered “large” in practice.