Quickest Real-Time Detection of Multiple Brownian Drifts

Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1832-1856, June 2024.
Abstract. Consider the motion of a Brownian particle in [math] dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, exactly [math] of the coordinate processes get a (known) nonzero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which the [math] coordinate processes get the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion without drift. The solution is expressed in terms of a stopping time that minimizes the probability of a false early detection and the expected delay of a missed late detection. The elliptic case [math] has been settled in [] where the hypoelliptic case [math] resolved in the present paper was left open (the case [math] reduces to the classic case [math] having a known solution). We also show that the methodology developed solves the problem in the general case where exactly [math] is relaxed to any number of the coordinate processes getting the drift. To our knowledge this is the first time that such a multidimensional hypoelliptic problem has been solved exactly in the literature.