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

## TL;DR

This paper derives a closed-form formula for the fractional moments of the quasi-stationary distribution of the Shiryaev diffusion on an interval, revealing its asymptotic behavior as the interval length grows.

## Contribution

It provides the first analytical expression for fractional moments of the distribution for any real order, extending previous results for integer moments.

## Key findings

- Fractional moments are explicitly derived for all real orders.
- As A approaches infinity, the fractional moments converge to those of an exponential distribution.
- The limiting distribution as A increases is the stationary distribution of the reciprocal process.

## 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 {\em arbitrary} given order $s\in\mathbb{R}$; the formula is consistent with that previously found by Polunchenko and Pepelyshev (2018) for the case of $s\in\mathbb{N}$. 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 $A\to+\infty$; the limiting exponential distribution is the stationary distribution of the reciprocal of the Shiryaev diffusion.