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