On the convergence rate of the quasi- to stationary distribution for the Shiryaev-Roberts diffusion. Issue 2 (2nd April 2020) - Li, Kexuan; Polunchenko, Aleksey S.
Abstract: 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 (c.d.f.), Q A ( x ), to its stationary c.d.f., H ( x ), as A → + ∞, is no worse than O ( log ( 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 (2017b ).