Abstract:
We consider some sequential analysis problems (to be more specific: sequential testing problems) and discuss results about a first-hitting time for a Brownian motion with a change-point. The first chapter of the Thesis deals with a class of stationary Gauss-Markov processes. It is assumed that about the observable process there are two simple hypotheses to test. A slightly modified Wald's approach is used for this problem, namely: we minimize the Kullback-Leibler divergence given the error probabilities are fixed at some level. An asymptotically optimal decision rule is found. The second chapter is dedicated to a Bayesian sequential testing problem on the finite time horizon for a Brownian bridge process. It is assumed that the observable process is a Brownian bridge starting from zero and having an unknown pinning point with a Bernoulli prior distribution. Our goal here is to tell apart the two naturally arisen simple hypothesis about the process in question. An optimal decision rule minimizing the sum of average observation time and penalties for a wrong terminal decision is found. At last, the third chapter treats with a Brownian motion with a change-point, it being an exponentially distributed random variable. Here the first-hitting time is investigated: the analytical expressions for Laplace transforms, average time, probability density and probability to hit the boundary for a finite time are established.
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