On the Convergence Rate of the Quasi- to Stationary Distribution for the Shiryaev-Roberts Diffusion

For the classical Shiryaev–Roberts martingale diffusion considered on the interval $[0,A]$, where $A>0$ is a given absorbing boundary, it is shown that the rate of convergence of the diffusion’s quasi-stationary cumulative distribution function (cdf), $Q_A(x)$, to its stationary cdf, $H(x)$, as $A\to +\mathrm{\infty }$, is no worse than $O(\log (A)\mathrm{/}A)$, uniformly in $x\ge 0$. The result is established explicitly, by constructing new tight lower- and upper-bounds for $Q_A(x)$ using certain latest monotonicity properties of the modified Bessel $K$ function involved in the exact closed-form formula for $Q_A(x)$ recently obtained by Polunchenko (2017).