The disorder problem for diffusion processes with the ε-linear and expected total miss criteria

We study the disorder problem for a time-homogeneous diffusion process. The aim is to determine an efficient detection strategy of the disorder time θ, at which the process changes its drift. We focus on the ϵ-linear and the expected total miss criteria, where, unlike the well known linear penalty criterion, the expected penalty for an early/wrong detection of θ is expressed as the frequency of false alarms launched at least ϵ units of time before θ and as the expected advance in the detection of θ, respectively. We show that the original optimal stopping problems can be reduced to a unifying optimal stopping problem; then, we derive the associated free-boundary problem and we provide sufficient conditions for the existence and uniqueness of its solution.