Introduction to the theory of diffusion processes. Transl. from the Russian by V. Khidekel a. G. Pasechnik.

*(English)*Zbl 0844.60050
Translations of Mathematical Monographs. 142. Providence, RI: AMS. xi, 271 p. (1995).

There are many excellent books on the market today dealing with stochastic processes in general, and with diffusion processes in particular. So why this English translation of N. V. Krylov’s ‘Introduction to the theory of diffusion processes’, published by the American Mathematical Society? In short, the approach taken in this book is quite different, not with respect to the material presented, but with respect to its style and to the line of development of the main ideas.

The concept is rather classical: Chapter 1 presents some basic concepts of probability theory, Chapter 2 discusses the Wiener process, with brief excursions into martingales and the Laplace operator. Chapter 3 introduces Itô’s integral with respect to the Wiener processes (and briefly with respect to the local martingales) and Itô’s formula. Chapter 4 contains some consequences, such as random time change, Girsanov’s theorem, and Burkholder-Davis-Gundy type inequalities. Chapter 5 develops the basic theory of stochastic differential equation, with emphasis on Kolmogorov’s forward and backward equations. Only the final Chapter 6 covers some nonstandard material, namely some smoothness results for the solutions of partial differential equations obtained via probabilistic methods.

What makes this book different, is the presentation of the material. The author starts from scratch, introducing all the necessary concepts and techniques as he needs them. This makes it easy to follow his line of thought and to get to the main topics, stochastic integrals and stochastic differential equations, without detour and without many prerequisites. Only a standard course in Advanced Calculus and some idea of the Lebesgue measure in \(\mathbb{R}\) are needed. Another invaluable help when studying from this book is a ‘dual’ presentation of the material: All the main concepts and results are accompanied by a discussion of the intuitive idea behind them, and almost all proofs are given in a straightforward and precise manner. Some easier proofs, or parts thereof, are formulated as exercises, often with useful hints. Actually, there is a third layer, called problems. These give extensions and additional related topics that are not used in the main body.

Of course, developing from scratch the basic theory of diffusion processes on 220 pages requires some restraint. The reader will not find topics like the law of large numbers, the central limit theorem, or conditional probabilities – the latter topic appears only in an intuitive way in the last, special chapter. Other concepts are introduced in a rather informal way, such as densities, or functions of bounded variation. This, together with the lack of Lebesgue-Stieltjes integrals, makes it difficult to understand the explanation why Wiener processes do not allow for ordinary integration. The style of presenting topics as needed leads to some interesting consequences: Results on martingales are scattered over Chapters 2-5, convergence in probability appears after the main results on the Itô integral, Wald distributions are treated after the introduction of the Wiener process, etc. While all this makes for interesting reading in an almost linear fashion, it also makes it quite hard to use the book as a reference, as it is difficult to find many results in their systematic context. The very short index aggravates the situation, not mentioning e.g. convergence in probability, Wald distribution, Ornstein-Uhlenbeck process, etc.

I have tried this book on several of my colleagues, who always wanted to know about diffusion processes, but with out lengthy excursion into systematic coverage of basic stochastic processes. The response was, in general, very favorable, mostly because they could see how various techniques of stochastic calculus were used to present stochastic differential equations in a straighforward manner. I would not hesitate to use this volume for an introductory course on diffusion processes for a mixed audience of science, engineering, economics, statistics, and mathematics students. However, for a special course in mathematics/statistics departments I would prefer a more structured and encyclopedic approach. And anyone interested in some applications of the theory, such as optimal control, filtering, or qualitative behavior will need to consult other literature.

The concept is rather classical: Chapter 1 presents some basic concepts of probability theory, Chapter 2 discusses the Wiener process, with brief excursions into martingales and the Laplace operator. Chapter 3 introduces Itô’s integral with respect to the Wiener processes (and briefly with respect to the local martingales) and Itô’s formula. Chapter 4 contains some consequences, such as random time change, Girsanov’s theorem, and Burkholder-Davis-Gundy type inequalities. Chapter 5 develops the basic theory of stochastic differential equation, with emphasis on Kolmogorov’s forward and backward equations. Only the final Chapter 6 covers some nonstandard material, namely some smoothness results for the solutions of partial differential equations obtained via probabilistic methods.

What makes this book different, is the presentation of the material. The author starts from scratch, introducing all the necessary concepts and techniques as he needs them. This makes it easy to follow his line of thought and to get to the main topics, stochastic integrals and stochastic differential equations, without detour and without many prerequisites. Only a standard course in Advanced Calculus and some idea of the Lebesgue measure in \(\mathbb{R}\) are needed. Another invaluable help when studying from this book is a ‘dual’ presentation of the material: All the main concepts and results are accompanied by a discussion of the intuitive idea behind them, and almost all proofs are given in a straightforward and precise manner. Some easier proofs, or parts thereof, are formulated as exercises, often with useful hints. Actually, there is a third layer, called problems. These give extensions and additional related topics that are not used in the main body.

Of course, developing from scratch the basic theory of diffusion processes on 220 pages requires some restraint. The reader will not find topics like the law of large numbers, the central limit theorem, or conditional probabilities – the latter topic appears only in an intuitive way in the last, special chapter. Other concepts are introduced in a rather informal way, such as densities, or functions of bounded variation. This, together with the lack of Lebesgue-Stieltjes integrals, makes it difficult to understand the explanation why Wiener processes do not allow for ordinary integration. The style of presenting topics as needed leads to some interesting consequences: Results on martingales are scattered over Chapters 2-5, convergence in probability appears after the main results on the Itô integral, Wald distributions are treated after the introduction of the Wiener process, etc. While all this makes for interesting reading in an almost linear fashion, it also makes it quite hard to use the book as a reference, as it is difficult to find many results in their systematic context. The very short index aggravates the situation, not mentioning e.g. convergence in probability, Wald distribution, Ornstein-Uhlenbeck process, etc.

I have tried this book on several of my colleagues, who always wanted to know about diffusion processes, but with out lengthy excursion into systematic coverage of basic stochastic processes. The response was, in general, very favorable, mostly because they could see how various techniques of stochastic calculus were used to present stochastic differential equations in a straighforward manner. I would not hesitate to use this volume for an introductory course on diffusion processes for a mixed audience of science, engineering, economics, statistics, and mathematics students. However, for a special course in mathematics/statistics departments I would prefer a more structured and encyclopedic approach. And anyone interested in some applications of the theory, such as optimal control, filtering, or qualitative behavior will need to consult other literature.

Reviewer: W.Kliemann (Ames)