Quickest Detection Problems for Ornstein-Uhlenbeck Processes
成果类型：
Article
署名作者：
Glover, Kristoffer; Peskir, Goran
署名单位：
University of Technology Sydney; University of Manchester
刊物名称：
MATHEMATICS OF OPERATIONS RESEARCH
ISSN/ISSBN：
0364-765X
发表日期：
2024
页码：
1045-1064
关键词：
摘要：
Consider an Ornstein-Uhlenbeck process that initially reverts to zero at a known mean-reversion rate beta 0, and then after some random/unobservable time, this meanreversion rate is changed to beta 1. Assuming that the process is observed in real time, the problem is to detect when exactly this change occurs 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 process prior to the change of its meanreversion rate. 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. Allowing for both positive and negative values of beta 0 and beta 1 (including zero), the problem and its solution embed many intuitive and practically interesting cases. For example, the detection of a mean-reverting process changing to a simple Brownian motion (beta 0 0 and beta 1 = 0) and vice versa (beta 0 = 0 and beta 1 > 0) finds a natural application to pairs trading in finance. The formulation also allows for the detection of a transient process becoming recurrent (beta 0 < 0 and 1 >= 0) as well as a recurrent process becoming transient (beta 0 >= 0 and beta 1 < 0). The resulting optimal stopping problem is inherently two-dimensional (because of a statedependent signal-to-noise ratio), and various properties of its solution are established. In particular, we find the somewhat surprising fact that the optimal stopping boundary is an increasing function of the modulus of the observed process for all values of 0 and beta 1.