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Martingales and stochastic analysis. (English) Zbl 0848.60001

Series on Multivariate Analysis. 1. Singapore: World Scientific. xiii, 501 p. (1995).
The book is a very high level text on the theory of stochastic processes, martingales, stochastic integration and stochastic differential equations. It mainly deals with the fundamental aspects of the theory, with extreme care, providing complete proofs of all the statements. It may be a very useful text for graduate and Ph. D. students, as well as a basic reference book for specialists.
The book consists of four chapters and an appendix. Chapter 1 treats basic topics of the theory of stochastic processes, including extensive discussion of \( \sigma\)-algebras, progressively measurable and predictable processes, approximation by simple processes, stopping times and truncation of processes, uniform integrability. Chapter 2 presents the foundations of the theory of martingales in great detail and deepness. The chapter includes the fundamental martingale inequalities, optional stopping theorems, optional sampling theorems, martingale convergence theorems, closing a martingale by a final element, uniform integrability, increasing processes as martingales and integrators and the Doob-Meyer decomposition theorem. Chapter 3 provides the basic definitions and theorems on the general theory of stochastic integration, in the framework of martingale theory, with predictable integrands and local \(L_2\)-martingales as integrators. Itô formula and Itô calculus are developed. Chapter 4 is an introduction to a general class of stochastic differential equations. The chapter treats in great detail basic facts on the space of continuous functions and function space representation of solutions, uniqueness in law and pathwise uniqueness, existence and uniqueness of strong solutions. Finally, a detailed appendix on stochastic independence, conditional expectation, regular conditional probabilities and multidimensional normal distributions is given.
Reviewer: F.Flandoli (Pisa)

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
60Hxx Stochastic analysis
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