Continuous exponential martingales and BMO / Norihiko Kazamaki

In order to report on more than two decades of constant progress in the theory of martingale inequalities, several authors have recently felt the necessity of summarizing its main features. This has been done recently by R. L. Long [Martingale spaces and inequalities, Peking Univ. Press, Beijing, 1993; MR1224450 (94j:60087)] and Kazamaki has independently written this short monograph. Some material is common to both books, such as the John-Nirenberg, Fefferman, reverse Hölder, Gehring, and weighted norm inequalities. But whereas Long's book was focused on the situation of discrete indices, the book under review entirely deals with continuous time local martingales with continuous paths. An essential role is played by exponential martingales and their connection with BMO martingales. Both are extensively studied in this continuous setting, and many examples or counterexamples illustrate the theoretical statements: many of them are gathered from the papers of the author or his students on the topic. Finally, this is a well-written small book which should be quite helpful in updating the information on this still topical subject. (MathSciNet)