General theory of stochastic processes under “usual conditions”

Abstract

Chapter 10General theory of stochastic processes is devoted to a systematic exposition of a continuous time version of stochastic analysis under “usual conditions” with its standard notions like a stochastic basis, filtration, stopping times, random sets, predictable and optional sigma-algebras etc. It is shown how the discrete time martingale theory as well as a pure continuous time theory of diffusion processes are generalized for so-called cadlag processes. Using the predictable notion of a compensator the fundamental Doob-Meyer theorem is formulated for the class of sub- and supermartingales ofClass D class D. The full version of stochastic integration of predictable processes with respect to square-integrable martingale is developed. Moreover, different decompositions of such martingales are proved as well as the Kunuta-Watanabe inequality. It is shown how the theory can be extended with the help of localization procedures (local martingales, processes with locally integrable variation, semimartingales). The Ito formula is proved for semimartingales. SDEs with respect to semimartingales are studied including the existence and uniqueness of solutions of such equations with the Lipschitz coefficients (see [2], [8], [9], [16], [18], [20], [26], [33], [36], and [37]).