Optimal real-time detection of a drifting Brownian coordinate. (English) Zbl 1476.60078
In this paper, the original problem of recognizing the component of a three-dimensional Brownian motion with a (known) non-zero drift coefficient under the assumption that two components have zero drift is analyzed. Based on the position of the Brownian particle observed in real time, the problem is to detect as soon as possible and with minimal probabilities of the wrong terminal decisions, which spatial coordinate has the nonzero drift. The problem is solved in the Bayesian formulation, under any prior probabilities of the nonzero drift being in any of the three spatial coordinates, when the passage of time, is penalized linearly. The main contribution of the article is finding the exact solution to the problem in three dimensions, including a rigorous treatment of its non-monotone optimal stopping boundaries.
Nota bene: The estimation of the drift has been considered by N. Privault and A. Réveillac [Ann. Stat. 36, No. 5, 2531–2550 (2008; Zbl 1274.62256)].
Nota bene: The estimation of the drift has been considered by N. Privault and A. Réveillac [Ann. Stat. 36, No. 5, 2531–2550 (2008; Zbl 1274.62256)].
Reviewer: Krzysztof J. Szajowski (Wrocław)
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:
optimal detection; sequential testing; Brownian motion; optimal stopping; elliptic partial differential equation; free-boundary problem; non-monotone boundary; smooth fit; nonlinear Fredholm integral equation; the change-of-variable formula with local time on surfacesCitations:
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