Non-additive disorder problems for some diffusion processes

We study the Bayesian problem of detecting a change in the drift rate of an observable diffusion process with certain non-additive detection delay penalty criterions. We express the Bayesian risk function through the current state of a multi-dimensional diffusion process playing the role of a Markovian sufficient statistic. In the case of exponential delay penalty costs, the optimal time of alarm is found as the first time at which the weighted likelihood ratio hits a stochastic boundary depending on the current observations. The proof is based on an embedding of the initial problem into an appropriate multi-dimensional optimal stopping problem and the analysis of the associated parabolic-type free-boundary problem. We provide closed form estimates for the value function and the boundary for that case, under certain nontrivial relations between the coefficients of the observable diffusion.