Quickest Detection Problems for Ornstein–Uhlenbeck Processes

Consider an Ornstein–Uhlenbeck process that initially reverts to zero at a known mean-reversion rate β 0 , and then after some random/unobservable time, this mean-reversion rate is changed to β 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 mean-reversion 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 β 0 and β 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 ( β 0 > 0 and β 1 = 0 ) and vice versa ( β 0 = 0 and β 1 > 0 ) finds a natural application to pairs trading in finance. The formulation also allows for the detection of a transient process becoming recurrent ( β 0 < 0 and β 1 ≥ 0 ) as well as a recurrent process becoming transient ( β 0 ≥ 0 and β 1 < 0 ). The resulting optimal stopping problem is inherently two-dimensional (because of a state-dependent 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 β 0 and β 1 .