Exact distribution of the Generalized Shiryaev–Roberts stopping time under the minimax Brownian motion setup

ABSTRACT

We consider the quickest change-point detection problem where the aim is to detect the onset of a prespecified drift in “live”-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The object of interest is the distribution of the stopping time associated with the Generalized Shryaev–Roberts (GSR) detection procedure set up to “sense” the presence of the drift in the Brownian motion under surveillance. Specifically, we seek the GSR stopping time's survival function (the tail probability that no alarm is triggered by the GSR procedure prior to a given point in time), and distinguish two scenarios: (a) when the drift never sets in (prechange regime) and (b) when the drift is in effect ab initio (postchange regime). Under each scenario, we obtain a closed-form formula for the respective survival function, with the GSR statistic's (deterministic) nonnegative headstart assumed arbitrarily given. The two formulae are 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. We then exploit the obtained formulae numerically and characterize the pre- and postchange distributions of the GSR stopping time depending on three factors: (1) magnitude of the drift, (2) detection threshold, and (3) the GSR statistic's headstart.

KEYWORDS:

• First passage times
• Generalized Shiryaev–Roberts procedure
• Kolmogorov forward equation
• Markov diffusion processes
• method of separation of variables
• parabolic partial differential equations
• quickest change-point detection
• sequential analysis
• Sturm–Liouville theory

MATHEMATICS SUBJECT CLASSIFICATION:

• 62L10
• 60G40
• 60J25
• 60J60
• 35K20

Acknowledgements

The author is thankful to the Editor-in-Chief, Nitis Mukhopadhyay (University of Connecticut–Storrs), and to the anonymous referee, whose constructive feedback provided on the first draft of the article helped improve the quality of the article and shape its current form. The author is also grateful to Dr. Grigory Sokolov (SUNY Binghamton) for the assistance provided with Mathematica.