__From the reviews:__ "... Both the Markov-process approach and the Itô approach ... have been immensely successful in diffusion theory. The Stroock-Varadhan book, developed from the historic 1969 papers by its authors, presents the martingale-problem approach as a more powerful - and, in certain regards, more intrinsic-means of studying the foundations of the subject. [...] ... the authors make the uncompromising decision not "to proselytise by intimidating the reader with myriad examples demonstrating the full scope of the techniques", but rather to persuade the reader "with a careful treatment of just one problem to which they apply". [...] Most of the main tools of stochastic-processes theory are used, ..but it is the formidable combination of probability theory with analysis ... which is the core of the work. [...] I have emphasized the great importance of the Stroock-Varadhan book. It contains a lot more than I have indicated; in particular, its many exercises conain much interesting material. For immediate confirmation of the subject’s sparkle, virtuosity, and depth, see ... McKean (‘s 1969 book). The Stroock-Varadhan book proceeds on its inexorable way like a massive Bach fugue. ... But old J.S. can e something of knockout if his themes get hold of you. And his influence on what followed was 8you may say) substantial." __David Williams__ in __the Bulletin of the American Mathematical Society__ Front Matter....Pages N1-XII Introduction....Pages 1-6 Preliminary Material: Extension Theorems, Martingales, and Compactness....Pages 7-45 Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure....Pages 46-64 Parabolic Partial Differential Equations....Pages 65-81 The Stochastic Calculus of Diffusion Theory....Pages 82-121 Stochastic Differential Equations....Pages 122-135 The Martingale Formulation....Pages 136-170 Uniqueness....Pages 171-194 Itô’s Uniqueness and Uniqueness to the Martingale Problem....Pages 195-207 Some Estimates on the Transition Probability Functions....Pages 208-247 Explosion....Pages 248-260 Limit Theorems....Pages 261-284 The Non-unique Case....Pages 285-303 Back Matter....Pages 304-338 "This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. This approach was initiated by Stroock and Varadhan in their famous papers. (...) The proofs and techniques are presented in such a way that an adaptation in other contexts can be easily done. (...) The reader must be familiar with standard probability theory and measure theory which are summarized at the beginning of the book. This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view." Mathematische Operationsforschung und Statistik, 1981 From the reviews: "This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. [...] This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view." Mathematische Operationsforschung und Statistik
