At the end of 1960s and the beginning of 1970s, when the Russian version
of this book was written, the 'general theory of random processes' did
not operate widely with such notions as semimartingale, stochastic
integral with respect to semimartingale, the Ito formula for
semimartingales, etc. At that time in stochastic calculus (theory of
martingales), the main object was the square integrable martingale. In a
short time, this theory was applied to such areas as nonlinear
filtering, optimal stochastic control, statistics for diffusion- type
processes. In the first edition of these volumes, the stochastic
calculus, based on square integrable martingale theory, was presented in
detail with the proof of the Doob-Meyer decomposition for submartingales
and the description of a structure for stochastic integrals. In the
first volume ('General Theory') these results were used for a
presentation of further important facts such as the Girsanov theorem and
its generalizations, theorems on the innovation pro- cesses, structure
of the densities (Radon-Nikodym derivatives) for absolutely continuous
measures being distributions of diffusion and ItO-type processes, and
existence theorems for weak and strong solutions of stochastic
differential equations. All the results and facts mentioned above have
played a key role in the derivation of 'general equations' for nonlinear
filtering, prediction, and smoothing of random processes.
