Title: Problems of Sequential Analysis for Natural Exponential Families

Abstract:

In the talk we will consider the following two problems of statistical
sequential analysis for natural exponential families of Levy processes
in both Bayesian and variational formulations: the Wald s sequential
hypotheses testing problem and the Kolmogorov-Shiryaev s problem of
quickest disorder (change-point) detection.  The Bayesian problems
reduce to optimal stopping problems for the relevant a posteriori
probability processes, which are proved to be Markovian. In their
order, the optimal stopping problems reduce to the correspondent
integro-differential free-boundary Stephan problems.  We give
sufficient conditions for solutions of free-boundary problems to be
solutions of the initial Bayesian problems in rater general case of
natural exponential families.  The key point of the talk will be the
presentation of explicit solutions of both Bayesian problems for
compound Poisson processes having exponentially distributed
jumps. These solutions are obtained by use of the principles of
continuous and the smooth fit for the value functions at the optimal
boundaries.  Also we determine the explicit stopping boundaries for
the Wald s sequential probability ratio test which is optimal in the
variational formulation of the sequential testing problem and remark
some facts about the solution of variational disorder problem for the
case of compound Poisson process with exponential jumps.