Analytic Evaluation of the Fractional Moments for the Quasi-Stationary Distribution of the Shiryaev Martingale on an Interval

04/05/2019

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by   Kexuan Li, et al.

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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.