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.