Asymptotic near-minimaxity of the randomized Shiryaev-Roberts-Pollak change-point detection procedure in continuous time

Abstract

Abstract. For the classical continuous-time quickest change-point detection problem it is shown that the randomized Shiryaev-Roberts-Pollak procedure is asymptotically nearly minimax-optimal (in the sense of Pollak [14]) in the class of randomized procedures with vanishingly small false alarm risk. The proof is explicit in that all of the relevant performance characteristics are found analytically and in a closed form. The rate of convergence to the (unknown) optimum is elucidated as well. The obtained optimality result is a one-order improvement of that previously obtained by Burnaev et al. [4] for the very same problem.

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

“…For example, from (15) and the trivial observation that M n > 0 for all n we readily obtain [25]. For applications of this result in quickest changepoint detection see [24,23]. Similarly, since the quasi-stationary distribution is supported on the interval [0, A], we may further deduce that M n ≤ A i M n−i for any i ∈ {0, 1, .…”

“…is a well-defined, valid particular solution to equation (29); note the similarity of ℓ (p) (u) to the right-hand side of identity (24).…”

“…Shiryaev-hence the name-in his fundamental work [29,30] on quickest change-point detection. While interest to the Shiryaev process in the context of quickest change-point detection has never weakened (see, e.g., [20,31,10,5,22,25,23,24]), the process has received a great deal of attention in other areas as well, notably in mathematical finance (see, e.g., [12,8,15]) and in mathematical physics (see, e.g., [18,7]). It has also been considered in the literature on general stochastic processes (see, e.g., [35,36,8,9,28,19,27,26]).…”

A Note on the Quasi-stationary Distribution of the Shiryaev Martingale on the Positive Half-Line

“…We can now conclude that, if A ∈ (0, A * ), then the quasi-stationary distribution's pdf and cdf are given by (30) and (31), respectively, with λ A ∈ (0, 1/8) determined as the only solution of equation (32).…”

“…Shiryaev in his fundamental work (see [34,35]) in the area of quickest change-point detection where the process (X t ) t 0 has become known as the Shiryaev-Roberts detection statistic, and it is one of the "central threads" in the field. See also, e.g., [29,36,14,3,31,30]. The second reason is that the time-homogeneous Markov diffusion (X t ) t 0 is easy to see to have the martingale property E[X t − x 0 − t] = 0 for any t 0, i.e., the process {X t − x 0 − t} t 0 is a zero-mean martingale.…”