Elements of classical theory of stochastic processes

Abstract

This chapter contains a general notion of random processes with continuous time. It is given in context of the Kolmogorov consistency theorem. The notion of a Wiener process with variety of its properties are also presented here. Its existence is stated by two ways: with the help of the Kolmogorov theorem as well as with the help of orthogonal functional systems. Besides the Wiener process as a basic process for many others, the Poisson process is also considered here. Stochastic integration with respect to Wiener process is developed for a class of progressively measurable functions. It leads to the Ito processes, the Ito formula, the Girsanov theorem and representation of martingales (see [5], [6], [14], [17], [21], [35], [41], and [44]).