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Stochastic integration and differential equations. 2nd ed. (English) Zbl 1041.60005
Applications of Mathematics 21. Berlin: Springer (ISBN 3-540-00313-4/hbk; 978-3-642-05560-7/pbk). xiii, 415 p. (2004).
In the new edition several changes have been made, most of them are inclusions of results achieved since the appearance of the first edition (1990; Zbl 0694.60047). In particular, Chapter III, “Semimartingales and decomposable processes”, has been rewritten. Now the notion of predictability and R. Bass’ proof of the Doob-Meyer decomposition theorem are used where before the presentation had been based on Meyer’s notion of natural processes. A section on compensators is added. In Chapter IV the treatment of martingale representation now includes the Jacod-Yor theorem and a section on sigma martingales. Chapter V has a new section, entitled “Eclectic useful results on stochastic differential equations”. Chapter VI, “Expansions of filtrations”, is altogether new. Another addition are exercises, to be found at the end of each chapter. Altogether I agree with the previous reviewer, the book provides an excellent basis for lecturing or self-teaching.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H05 Stochastic integrals
60G07 General theory of stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)