Sequential bayesian and minimax decisions based on stochastic processes

Abstract

For a large class of sequential decision problems in continuous time the problem of finding an optimal sequential decision procedure is reduced to an optimal stopping problem for a certain strochastic process (X_t). Conditions with respect to the basic data are given such that the process has continuity properties which yield the existence of an optimal stopping time. In case, in which the terminal decision is based on the observation of a stochastic process (e.g. Gaussian process, semimartingale with Gaussian martingale part, Gaussian Markovian process), a representation of (X_t) can be derived if conjugate priors and appropriate but rather general loss functions are used. In special cases explicit solutions of the optimal stopping problems can be found, too.

Keywords:

• sequential analysis in continuous time
• Bayes procedures
• statistics of stochastic processes
• Gaussian processes
• stopping times
• stochastic integrals