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Ernst, Philip A.; Peskir, Goran

Quickest real-time detection of a Brownian coordinate drift. (English) Zbl 1499.60129

Ann. Appl. Probab. 32, No. 4, 2652-2670 (2022).
Summary: Consider the motion of a Brownian particle in two or more dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, one of the coordinate processes gets 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 a coordinate process gets 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 minimises the probability of a false early detection and the expected delay of a missed late detection. To our knowledge this is the first time that such a problem has been solved exactly in the literature.

Cited in 1 Review
Cited in 2 Documents

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60J65 Brownian motion
60H30 Applications of stochastic analysis (to PDEs, etc.)
35J15 Second-order elliptic equations
45G10 Other nonlinear integral equations
62C10 Bayesian problems; characterization of Bayes procedures

Keywords:

Brownian motion; elliptic partial differential equation; free-boundary problem; nonlinear Fredholm integral equation; optimal stopping; quickest detection; smooth fit; the change-of-variable formula with local time on surfaces
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References:

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