On the Quasi-Stationary Distribution of the Shiryaev--Roberts Diffusion

We consider the diffusion $(R_t^r)_{t\ge0}$ generated by the equation $dR_t^r=dt+\mu R_t^r dB_t$ with $R_0^r\triangleq r\ge0$ fixed, and where $\mu\neq0$ is given, and $(B_t)_{t\ge0}$ is standard Brownian motion. We assume that $(R_t^r)_{t\ge0}$ is stopped at $\mathcal{S}_A^r\triangleq\inf\{t\ge0\colon R_t^r=A\}$ with $A>0$ preset, and obtain a closed-from formula for the quasi-stationary distribution of $(R_t^r)_{t\ge0}$, i.e., the limit $Q_A(x)\triangleq\lim_{t\to+\infty}\Pr(R_t^r\le x|\mathcal{S}_A^r>t)$, $x\in[0,A]$. We also prove $Q_A(x)$ to be unimodal for any $A>0$, and obtain its entire moment series. More importantly, the pair $(\mathcal{S}_A^r,R_t^r)$ with $r\ge0$ and $A>0$ is the well-known Generalized Shiryaev--Roberts change-point detection procedure, and its characteristics for $r\sim Q_A(x)$ are of particular interest, especially when $A>0$ is large. Driven by this circumstance, we offer an order-three large-$A$ asymptotic approximation of $Q_A(x)$ valid for all $x\in[0,A]$. The approximation is rather accurate even if $A$ is lower than what would be considered "large" in practice.