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 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 the formula is consistent with that previously found by Polunchenko and Pepelyshev for the case of 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 the limiting exponential distribution is the stationary distribution of the reciprocal of the Shiryaev diffusion.