Analytic moment and Laplace transform formulae for the quasi-stationary distribution of the Shiryaev diffusion on an interval

Abstract

We derive analytic closed-form moment and Laplace transform formulae for the quasi-stationary distribution of the classical Shiryaev diffusion restricted to the interval [0, A] with absorption at a given A > 0.Keywords Laplace transform · Markov diffusions · Quasi-stationarity · Shiryaev process · Special functions · Stochastic processes Mathematics Subject Classification (2000) 60J60 · 60J25 IntroductionThis work is an investigation into quasi-stationarity of the classical Shiryaev diffusion restricted to an interval. Specifically, the focus is on the solution (R r t ) t≥0 of the stochastic differential equationwhere (B t ) t≥0 is standard Brownian motion in the sense that E[dB t ] = 0, E[(dB t ) 2 ] = dt, and B 0 = 0. The time-homogeneous Markov process (R r t ) t≥0 is an important particular versionThe effort of A.S.

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

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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.

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“…Lemma 4.1 (Polunchenko and Pepelyshev 2018). For every A > 0 fixed, the solution {M n } n 0 to the recurrence (14) is given by…”

“…The Q-to-H convergence was recently investigated further by Li and Polunchenko (2019) who used the explicit formula obtained by Polunchenko (2017a) for Q A (x) to demonstrate that the uniform (in x) rate of convergence of Q A (x) down to H(x) as A → +∞ is no slower than O(log(A)/A). While the stationary distribution (3)-(4) is momentless (no moments of orders one and higher, but moments of orders s < 1 do exist), the quasi-stationary distribution (2) is not: its entire (positiveinteger-order) moment series was recently evaluated explicitly by Polunchenko and Pepelyshev (2018) through the use of the closed-form formulae for Q A (x) and q A (x) obtained earlier by Polunchenko (2017a). The contribution of this work is an explicit formula (in different equivalent forms) for the quasi-stationary distribution's fractional moment of any given order s ∈ R; the existence of the s-th order fractional moment is shown formally as well.…”

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

“…(Polunchenko, 2016, p. 136 and Lemma 3.3). See also Polunchenko and Pepelyshev (2018) for a discussion of potential ways to improve the foregoing double inequality.…”

“…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.…”