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

Aleksey S. Polunchenko () and Grigory Sokolov ()
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Aleksey S. Polunchenko: State University of New York at Binghamton
Grigory Sokolov: State University of New York at Binghamton

Methodology and Computing in Applied Probability, 2016, vol. 18, issue 4, 1153-1195

Abstract: 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.

Keywords: Generalized Shiryaev-Roberts procedure; Kolmogorov forward equation; Markov diffusion processes; Quickest change-point detection; Sequential analysis; MSC 62L10; MSC 60G10; MSC 62M15; MSC 60J60 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s11009-016-9478-7

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