  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 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 (pre-change regime) and (b)
when the drift is in effect ab initio (post-change 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
post-change distributions of the GSR stopping time depending on three factors:
(a) magnitude of the drift, (b) detection threshold, and (c) the GSR
statistic's headstart.
