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

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 [Ann. Statist., 13 (1985), pp. 206–227]) 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, Feinberg, and Shiryaev [Theory Probab. Appl., 53 (2009), pp. 519–536] for the very same problem.