Integration of vector-valued functions and optimal estimation of stochastic processes

In this section. we investigate the so-called prognosis problem which can be formulated as follows. Suppose that two stochastic processes X and Y are given, such that X is not accessible for a direct observation, but Y is completely accessible. Suppose also that there are some close relationships between these two processes. Then it is natural to estimate the values of X starting with the corresponding values of Y (in other words, having a certain information about the values of Y, we want to prognosticate the corresponding values of X). Problems of this type arise in various situations and play a significant role in mathematical statistics. In order to explain the deep connection of such problems with topics of our book, it is necessary to formulate these problems in a more precise manner and to specify the meaning of the phrase “prognostication”.