Elements of stochastic calculus and analysis
Professor Daniel Stroock, who is well known for his long standing contribution to modern probability theory, has written another original book. It is remarkable to see the convincing way of telling the readers about new aspects when writing on one of the classical topics, `stochastic calculus'. \par The main discussion is on variations of Kolmogorov's equations, both forward and backward, and their deep relation with stochastic processes of diffusion type. While this is basically the contents of Chapter 1, the author involves the Brownian motion process and the Wiener measure to present in Chapters 2 and 3 the Itô's approach and the construction of Itô's stochastic integration. Several results about paths properties of stochastic processes are discussed. Some results are interesting extensions of well-known results from the classical stochastic calculus. Topics like `Wiener's spaces of homogeneous chaos' are considered. In Chapter 4, the author writes on other theories of stochastic integration (Stratonovich, Kunita and Watanabe, Doob and Meyer). Finally, in Chapter 5 called `Addenda', we find a compact and comprehensive presentation of topics such as martingale problem, exponential semi-martingales, Feynman-Kac formula, Cameron-Martin-Girsanov formula, Kalman-Bucy filter and Malliavin calculus. \par At the end of each chapter there are challenging exercises. In order to be successful, the reader should work not only on the theory developed but also to make all details for the exercises. \par The style is compact, however the author provides a lot of details for practically any of the results presented in the book. Even readers who believe that are in the area of stochastic calculus will find either new results or a new view at known results. \par By writing this book the author has shown once again that he is one of the leading masters of modern probability theory. There are all the expectations to believe that the book will be met positively and will be useful and encouraging for both young mathematicians and professionals working in the areas of probability theory and its applications and analysis.
scientific article; zbMATH DE number 6856393