An Analytic Expression for the Distribution of the Generalized Shiryaev-Roberts Diffusion

TL;DR

This paper derives an explicit formula for the distribution of the GSR detection statistic in the pre-change regime of a Brownian motion, enabling better understanding of its behavior before a drift occurs.

Contribution

It provides a closed-form analytical expression for the transition probability density of the GSR statistic, extending known stationary distribution results.

Findings

Derived the transition pdf analytically using spectral methods.

Generalized the stationary distribution formula for the GSR statistic.

Numerically analyzed the pre-change behavior of the GSR statistic.

Abstract

We consider the quickest change-point detection problem where the aim is to detect the onset of a pre-specified drift in "live"-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The topic of interest is the distribution of the Generalized Shryaev-Roberts (GSR) detection statistic set up to "sense" the presence of the drift. Specifically, we derive a closed-form formula for the transition probability density function (pdf) of the time-homogeneous Markov diffusion process generated by the GSR statistic when the Brownian motion under surveillance is "drift-free", i.e., in the pre-change regime; the GSR statistic's (deterministic) nonnegative headstart is assumed arbitrarily given. The transition pdf formula is found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. The obtained result generalizes the well-known formula for the (pre-change) stationary distribution of the GSR statistic: the latter's stationary distribution is the temporal limit of the distribution sought in this work. To conclude, we exploit the obtained formula numerically and briefly study the pre-change behavior of the GSR statistic versus three factors: (a) drift-shift magnitude, (b) time, and (c) the GSR statistic's headstart.