Bayesian change-point detection for a Brownian motion with a total miss criterion

We study the following problem of sequential analysis: we observe a Brownian motion which has a zero drift initially; at a an unknown and random time θ, known as change-point, the Brownian motion takes a non-zero drift. Since the Brownian motion is observed in real time, we want to estimate θ optimally by means of a stopping time which minimizes a total miss criterion, namely the linear combination between the expected advance in detecting θ wrongly and expected delay of a late detection. This problem is solved in the Bayesian formulation, where θ is assumed to follow an exponential prior distribution.