On Markovian sufficient statistics in non-additive disorder problems for jump-diffusion processes

We study the Bayesian problems of quickest (online) detection of unknown changes in the probabilistic characteristics of continuously observable jump-diffusion processes with the non-additive detection delay penalty criteria proposed by Shiryaev [49]. The reward functionals of the Bayesian risks are expressed through the current states of the multidimensional jump-diffusion processes playing the role of Markovian sufficient statistics in the appropriate optimal stopping problems. We derive stochastic differential equations for the components of the resulting multi-dimensional Markov processes and discuss possible extensions of the results to the case of observable discontinuous processes with stationary and independent increments forming natural exponential families.