On Markov Sufficient Statistics in Nonadditive Bayes Problems of Sequential Analysis

The question of finding Markov sufficient statistics (see definition 3) in the problem of minimisation of the functional (2) is considered. It is supposed that the parameter $\theta$ the random moment at which the density $f_0$ changes to $f_1$ (§ 1–3) or to one of the $f_1,\dots,f_m$ (§ 5). In the case when the densities $f_0,f_1,\dots,f_m$ belong to the exponential family and the functional which is minimized is a non-additive one of a special form, we find a finite number of Markov sufficient statistics. Connections between the problem considered and other problems of sequential analysis are also discussed.