##
**Multidimensional diffusion processes.
Reprint of the 2nd correted printing (1997).**
*(English)*
Zbl 1103.60005

Classics in Mathematics. Berlin: Springer (ISBN 3-540-28998-4/hbk). xii, 338 p. (2006).

[The first edition (1979) has been reviewed in Zbl 0426.60069.]

The main purpose of this classic book is to elucidate the martingale approach to the theory of Markov processes and to propose a study of diffusion theory in \({\mathbb R}^d\).

The book is organized as follows. Chapter 1 provides an introduction to those parts of measure and probability theories which are considered most important for an understanding of the book. Basic tools are developed which are necessary for the construction of measures on functional spaces and introduces some notions and results which will play an important role in what follows. In Chapters 2 and 3 basic notions and results concerning Markov processes and parabolic partial differential equations are referred. A sketch of the procedure of studying diffusion theory via the backward equation is given which turns out to be one of the more powerful and successful approaches to the subject. In Chapter 4 Itô’s theory of stochastic integration is developed. In Chapter 5 this theory is applied for studying the equation \(dx(t)=\sigma(t, x(t)) d \beta(t) + b(t, x(t)) dt\) where \(\beta(\cdot)\) is a \(d\)-dimensional Brownian motion. In Chapter 6 the study of diffusion theory is begun and a basic existence theorem for solutions to the martingale problem is proved. The relationship between the martingale problem and the (strong) Markov property, as well as the formula of Cameron, Martin and Girsanov are shown. Chapter 7 contains a proof of the general theorem about uniqueness for the solution of martingale problem. Chapter 8 expands the investigation of the relationships between Itô’s approach and the martingale problem. The results of Chapter 9 are various \(L^p\)-estimates for the transition probability function of the considered multidimensional diffusion process. Chapter 10 extends the martingale problem approach to the case of unbounded coefficients. Standard conditions that can be used to test for explosion are given. In Chapter 11 the stability results for the Markov processes are studied. These results can be naturally divided into two categories: convergence of Markov chains to diffusions and convergence of diffusions to other diffusions. Chapter 12 takes up the question of what can be done in those circumstances when existence of solutions to a martingale problem can be proved but uniqueness cannot. It is also shown that every solution to a given martingale problem can, in some sense, be built out of those solutions which are part of a Markov family. In the Appendix some results from outside probability theory and concerning to the theory of singular integrals that are relied in Chapters 7 and 9 are proved in order to make the book self-contained.

The main purpose of this classic book is to elucidate the martingale approach to the theory of Markov processes and to propose a study of diffusion theory in \({\mathbb R}^d\).

The book is organized as follows. Chapter 1 provides an introduction to those parts of measure and probability theories which are considered most important for an understanding of the book. Basic tools are developed which are necessary for the construction of measures on functional spaces and introduces some notions and results which will play an important role in what follows. In Chapters 2 and 3 basic notions and results concerning Markov processes and parabolic partial differential equations are referred. A sketch of the procedure of studying diffusion theory via the backward equation is given which turns out to be one of the more powerful and successful approaches to the subject. In Chapter 4 Itô’s theory of stochastic integration is developed. In Chapter 5 this theory is applied for studying the equation \(dx(t)=\sigma(t, x(t)) d \beta(t) + b(t, x(t)) dt\) where \(\beta(\cdot)\) is a \(d\)-dimensional Brownian motion. In Chapter 6 the study of diffusion theory is begun and a basic existence theorem for solutions to the martingale problem is proved. The relationship between the martingale problem and the (strong) Markov property, as well as the formula of Cameron, Martin and Girsanov are shown. Chapter 7 contains a proof of the general theorem about uniqueness for the solution of martingale problem. Chapter 8 expands the investigation of the relationships between Itô’s approach and the martingale problem. The results of Chapter 9 are various \(L^p\)-estimates for the transition probability function of the considered multidimensional diffusion process. Chapter 10 extends the martingale problem approach to the case of unbounded coefficients. Standard conditions that can be used to test for explosion are given. In Chapter 11 the stability results for the Markov processes are studied. These results can be naturally divided into two categories: convergence of Markov chains to diffusions and convergence of diffusions to other diffusions. Chapter 12 takes up the question of what can be done in those circumstances when existence of solutions to a martingale problem can be proved but uniqueness cannot. It is also shown that every solution to a given martingale problem can, in some sense, be built out of those solutions which are part of a Markov family. In the Appendix some results from outside probability theory and concerning to the theory of singular integrals that are relied in Chapters 7 and 9 are proved in order to make the book self-contained.

Reviewer: Pavel Gapeev (Berlin)

### MathOverflow Questions:

What work has been done on SDE with diffusion coefficients of bounded variation in \(\mathbb R^d\)?### MSC:

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

60J60 | Diffusion processes |

60G44 | Martingales with continuous parameter |

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

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60F17 | Functional limit theorems; invariance principles |

93Exx | Stochastic systems and control |