Report
Quickest Real-Time Detection of Multiple Brownian Drifts
| العنوان: | Quickest Real-Time Detection of Multiple Brownian Drifts |
|---|---|
| المؤلفون: | Ernst, Philip A., Mei, Hongwei, Peskir, Goran |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, 60G40, 60J65, 60H30 |
| الوصف: | Consider the motion of a Brownian particle in $n$ dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, exactly $k$ of the coordinate processes get a (known) non-zero 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 $k$ 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 minimises the probability of a false early detection and the expected delay of a missed late detection. The elliptic case $k=1$ has been settled in Ernst and Peskir (2022) where the hypoelliptic case $1 < k < n$ resolved in the present paper was left open (the case $k = n$ reduces to the classic case $n=1$ having a known solution). We also show that the methodology developed solves the problem in the general case where exactly $k$ is relaxed to any number of the coordinate processes getting the drift. To our knowledge this is the first time that such a multi-dimensional hypoelliptic problem has been solved exactly in the literature. Comment: 26 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2007.14786 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2305.05721 |
| رقم الانضمام: | edsarx.2305.05721 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |