Analytic evaluation of the fractional moments for the quasi-stationary distribution of the Shiryaev martingale on an interval

Abstract

We consider the quasi-stationary distribution of the classical Shiryaev diffusion restricted to the interval [0, A] with absorption at a fixed A > 0. We derive analytically a closed-form formula for the distribution's fractional moment of an arbitrary given order s ∈ R; the formula is consistent with that previously found by Polunchenko and Pepelyshev (2018) for the case of s ∈ N. We also show by virtue of the formula that, if s < 1, then the s-th fractional moment of the quasi-stationary distribution becomes that of the exponential distribution (with mean 1/2) in the limit as A → +∞; the limiting exponential distribution is the stationary distribution of the reciprocal of the Shiryaev diffusion.

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

“…Moreover, analytic closed-form formulae for Q A (x) and q A (x) were recently obtained by Polunchenko (2017c), apparently for the first time in the literature; see formulae (3.3) and (3.4) in Section 3 below. These formulae were used by Polunchenko and Pepelyshev (2018) to compute analytically the quasi-stationary distribution's Laplace transform, and then also by Li et al (2019) to find the quasi-stationary distribution's fractional moment of any real order.…”