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

07/05/2019

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by   Kexuan Li, et al.

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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→+∞, is no worse than O((A)/A), uniformly in x>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 (2016).