From the reviews to the first edition: Most of the literature about stochastic differential equations seems to place so much emphasis on rigor and completeness that it scares the nonexperts away. These notes are an attempt to approach the subject from the nonexpert point of view.: Not knowing anything … about a subject to start with, what would I like to know first of all. My answer would be: 1) In what situations does the subject arise ? 2) What are its essential features? 3) What are the applications and the connections to other fields?" The author, a lucid mind with a fine pedagocical instinct, has written a splendid text that achieves his aims set forward above. He starts out by stating six problems in the introduction in which stochastic differential equations play an essential role in the solution. Then, while developing stochastic calculus, he frequently returns to these problems and variants thereof and to many other problems to show how thetheory works and to motivate the next step in the theoretical development. Needless to say, he restricts himself to stochastic integration with respectto Brownian motion. He is not hesitant to give some basic results without proof in order to leave room for "some more basic applications"… It can be an ideal text for a graduate course, but it is also recommended to analysts (in particular, those working in differential equations and deterministic dynamical systems and control) who wish to learn quickly what stochastic differential equations are all about. From: Acta Scientiarum Mathematicarum, Tom 50, 3-4, 1986.

This text on stochastic processes and their applications is based on a set of lectures given during the past several years at the University of California, Santa Barbara (UCSB). It is an introductory graduate course designed for classroom purposes. Its objective is to provide graduate students of statistics with an overview of some basic methods and techniques in the theory of stochastic processes. The only prerequisites are some rudiments of measure and integration theory and an intermediate course in probability theory. There are more than 50 examples and applications and 243 problems and complements which appear at the end of each chapter. The book consists of 10 chapters. Basic concepts and definitions are pro­ vided in Chapter 1. This chapter also contains a number of motivating ex­ amples and applications illustrating the practical use of the concepts. The last five sections are devoted to topics such as separability, continuity, and measurability of random processes, which are discussed in some detail. The concept of a simple point process on R+ is introduced in Chapter 2. Using the coupling inequality and Le Cam's lemma, it is shown that if its counting function is stochastically continuous and has independent increments, the point process is Poisson. When the counting function is Markovian, the sequence of arrival times is also a Markov process. Some related topics such as independent thinning and marked point processes are also discussed. In the final section, an application of these results to flood modeling is presented.

This thesis makes significant advances in the use of microspheres in optical traps as highly precise sensing platforms. While optically trapped microspheres have recently proven their dominance in aqueous and vacuum environments, achieving state-of-the-art measurements of miniscule forces and torques, their sensitivity to perturbations in air has remained relatively unexplored. This thesis shows that, by uniquely operating in air and measuring its thermally-fluctuating instantaneous velocity, an optically trapped microsphere is an ultra-sensitive probe of both mass and sound. The mass of the microsphere is determined with similar accuracy to competitive methods but in a fraction of the measurement time and all while maintaining thermal equilibrium, unlike alternative methods. As an acoustic transducer, the air-based microsphere is uniquely sensitive to the velocity of sound, as opposed to the pressure measured by a traditional microphone.

The modem theory of Markov processes has its origins in the studies of A. A. MARKOV (1906-1907) on sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian motion (L. BACHELlER 1900, A. EIN­ STEIN 1905). The first correct mathematical construction of a Markov process with continuous trajectories was given by N. WIENER in 1923. (This process is often called the Wiener process.) The general theory of Markov processes was developed in the 1930's and 1940's by A. N. KOL­ MOGOROV, W. FELLER, W. DOEBLlN, P. LEVY, J. L. DOOB, and others. During the past ten years the theory of Markov processes has entered a new period of intensive development. The methods of the theory of semigroups of linear operators made possible further progress in the classification of Markov processes by their infinitesimal characteristics. The broad classes of Markov processes with continuous trajectories be­ came the main object of study.

Addressed to both pure and applied probabilitists, including graduate students, this text is a pedagogically-oriented introduction to the Schwartz-Meyer second-order geometry and its use in stochastic calculus. P.A. Meyer has contributed an appendix: "A short presentation of stochastic calculus" presenting the basis of stochastic calculus and thus making the book better accessible to non-probabilitists also. No prior knowledge of differential geometry is assumed of the reader: this is covered within the text to the extent.

This book—an outgrowth of an international conference held in honor of Jacques Peyrière—provides readers with an overview of recent developments in the mathematical fields related to fractals. Included are original research contributions as well as surveys written by experts in their respective fields.

With the first edition out of print, we decided to arrange for republi­ cation of Denumerrible Markov Ohains with additional bibliographic material. The new edition contains a section Additional Notes that indicates some of the developments in Markov chain theory over the last ten years.

Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. In a lively and imaginative presentation, studded with examples, exercises, and applications, and supported by inclusion of computational procedures, the author has created a textbook that provides easy access to this fundamental topic for many students of applied sciences at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is accessible to beginners but flexible enough to serve as well those who come to the course with strong backgrounds.

This new, thoroughly revised and expanded 3rd edition of a classic gives a comprehensive coverage of modern probability in a single book. It is a truly modern text, providing not only classical results but also material that will be important for future research. Much has been added to the previous edition, including eight entirely new chapters on subjects like random measures, Malliavin calculus, multivariate arrays, and stochastic differential geometry.

This research monograph develops the Hamilton-Jacobi-Bellman (HJB) theory through dynamic programming principle for a class of optimal control problems for stochastic hereditary differential systems. It is driven by a standard Brownian motion and with a bounded memory or an infinite but fading memory.

This book introduces data-driven remaining useful life prognosis techniques, and shows how to utilize the condition monitoring data to predict the remaining useful life of stochastic degrading systems and to schedule maintenance and logistics plans. It is also the first book that describes the basic data-driven remaining useful life prognosis theory systematically and in detail.

Let us begin by quoting from the Preface to the author's Statistical Physics (Moscow, Nauka 1982; also published in English by Harwood in 1986): '''My God! Yet another book on statistical physics! There's no room on my bookshelves left!' Such emotionsare quite understandable. Beforejumping to conclusions, however, it would be worthwhile to read the Introduction and look through the table of contents.

This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter.

Stochastic Tools in Mathematics and Science is an introductory book on probability-based modeling. It covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. The topics covered include conditional expectations, stochastic processes, Brownian motion and its relation to partial differential equations, Langevin equations, the Liouville and Fokker-Planck equations, as well as Markov chain Monte Carlo algorithms, renormalization and dimensional reduction, and basic equilibrium and non-equilibrium statistical mechanics. The applications include data assimilation, prediction from partial data, spectral analysis, and turbulence. A noteworthy feature of the book is the systematic analysis of memory effects.