Nonadditive disorder problems for some diffusion processes
- Gapeev, Pavel
2010 Mathematics Subject Classification
- 60G40 62M20 34K10 62C10 62L15 60J60
- Quickest 'disorder'(change-point) detection, diffusion process, optimal stopping, sufficient statistic, free-boundary problem, smooth-fit and normal-entrance conditions, Itô's formula
We study nonadditive Bayesian problems of detecting a change in drift of an observed diffusion process where the cost function of the detection delay has the same structure as in  and construct a finite-dimensional Markovian sufficient statistic for that case. We show that when the cost function is linear the optimal stopping time is found as the first time when the a posteriori probability process hits a stochastic boundary depending on the observation process. It is shown that under some nontrivial relationships on the coefficients of the observed diffusion the problem admits a closed form solution. The method of proof is based on embedding the initial problem into a two-dimensional optimal stopping problem and solving the equivalent free-boundary problem by means of the smooth-fit conditions.