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**Stochastic processes.**
*(English)*
Zbl 1133.60004

Courant Lecture Notes in Mathematics 16. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (ISBN 978-0-8218-4085-6/pbk). ix, 126 p. (2007).

This small volume of about \(120\) pages by S. R. S. Varadhan comprises an introduction to stochastic continuous time process on the real line. It is the continuation of a first volume on probability theory by the same author and is additionally based on a course given at the Courant Institute, New York. The matter is arranged in seven chapters of strongly varying size in the following way. First the author gives a brief introduction to semimartingales and an interesting glance about the convergence of random series as preliminary to stochastic integration carried out in detail in a later chapter. The first narrative line establishes jump processes along the pedagogic line from Poisson process to a general Lévy process culminating in the Lévy-Chinchine formula. Passing shortly the perspective of Poisson point processes he widens the field to Markov jump processes which he treats with some detail. All this is vividly illustrated by birth and death processes. The second main plot line starts with the construction of Brownian motion and the Wiener measure and the account of their main properties. In the sequel stochastic integration, the author establishes stochastic integration and proves the main theorems of stochastic analysis. This most voluminous chapter of the text ends with a chain of about five interesting but still smaller subsections. The following chapter deals with the standard theory of diffusions and explains the intuition behind the notion of weak solutions although never actually using this term. As concluding chapter the author collects all the parts developed so far and gives a very short introduction to general Markov processes and their main features. The appendices deliver helpful material about measures on Polish spaces and the spaces \(C([0,1])\) and \(D([0,1])\). This volume is written in an elegant, very intuitive and lucid style, which obviously relies on a long teaching experience. Wherever possible, technicialities are avoided in order to keep ideas clear and insightful. Proofs are carried out in an extremely efficient manner, mostly until a point where a graduate student should be able to complete the last missing line. This efficiency allows to give at least a detailed sketch of proof for every single statement on these comparable few pages. The wise composition of the text allows to give a wonderful overview over general Markov processes based on the phenomenologically illuminating treatment of different kinds of jump processes and diffusion processes. What enriches this volume with a particularly precious flavour is the recurrent leitmotiv of enlightening Markov processes via associated martingales. I strongly do recommend this textbook equally for academic teachers as for students. It is a reliable and inspiring starting ground in the theory of continuous time stochastic processes explained by one of the masters of the field and a bright example of rational reasoning itself.

Reviewer: Michael Högele (Berlin)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G05 | Foundations of stochastic processes |

60G07 | General theory of stochastic processes |

60G44 | Martingales with continuous parameter |

60G51 | Processes with independent increments; Lévy processes |

60J25 | Continuous-time Markov processes on general state spaces |