Chapter 9 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 and 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 of Class 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. Stochastic differential equations (SDEs) with respect to semimartingales are studied including the existence and uniqueness of solutions of such equations with the Lipschitz coefficients (see Beiglboeck et al., Stoch Process Appl 122(4):1204–1209, 2012; Cohen and Elliott, Stochastic calculus and applications, 2nd edn., 2015; Doléans–Dade, Stochastic processes and stochastic differential equations, pp. 7–73, 2010; Kallianpur and Karandikar, Introduction to option pricing theory, 2012; Klebaner, Introduction to stochastic calculus with applications, 2012; Kruglov, Stochastic processes, 2013; Liptser and Shiryaev, Theory of martingales, 1989; Meyer, Probability and potential, 1966; Protter, Stochastic integration and differential equations, 2nd edn., 2005, and Revuz and Yor, Continuous martingales and brownian motion, 2nd edn., 1999).