An intrinsic bound for the number of observations in an optimal sequential decision problem

We consider a problem of sequential statistical decisions for the case of unbounded procedures. Provided no external bound is supplied for the number of observations, the optimal procedure may be obtained by means of backwards induction only in the case an intrinsic upper bound exists and can be found. In the present paper, under some general hypotheses, we show that such intrinsic upper bound exists, and we provide a way to find it. Moreover, a method is given to reduce as much as possible the required amount of computation for the backwards induction. An immediate application of the proposed method may be seen in the field of Quality Control.