The theory of stochastic processes II. Transl. from the Russian by S. Kotz. Repr. of the 1st ed. 1975.

*(English)*Zbl 0531.60002
Grundlehren der Mathematischen Wissenschaften, Bd. 218. Berlin etc.: Springer-Verlag. VII, 441 p. DM 148.00; $ 63.80 (1983).

For a review of the Russian edition see Zbl 0298.60024 and for the first English edition Zbl 0305.60027.

Chap. I. Basic definitions and properties of Markov processes: properties of families of transition probabilities, strong Markov property, multiplicative functionals, sample path properties. Chap. II. Homogeneous Markov processes: Semi-group theory, Hille-Yosida theorem, Feller processes, additive functionals of Markov processes, random time substitution. Chap. III. Jump processes: General definitions, countable state spaces, semi-Markov processes, Markov processes with a discrete component. Chap. IV. Processes with independent increments: One- dimensional and multidimensional state space, first passage times, extremal values. Chap. V Markov branching processes: Finite number of particle types, continuous mass distribution, general Markov processes with branching. The book is closed by a bibliography and a short section with historical remarks.

This second volume presents a rather complete survey of the (now) classical theory of Markov processes following the treatment of such processes by Dynkin: Starting with measure theoretic definitions of Markov processes the associated operators are investigated. Excluded is the field of diffusion processes. The book is written on a mathematical rigorous level, presenting the theory without its applications. So a reader should have knowledge of the applications of Markov processes, and should be familiar with at least a part of the theory of stochastic processes. More or less it seems that the book is written for those who are specialists in the theory of Markov processes, or for those who want to become a specialist.

Chap. I. Basic definitions and properties of Markov processes: properties of families of transition probabilities, strong Markov property, multiplicative functionals, sample path properties. Chap. II. Homogeneous Markov processes: Semi-group theory, Hille-Yosida theorem, Feller processes, additive functionals of Markov processes, random time substitution. Chap. III. Jump processes: General definitions, countable state spaces, semi-Markov processes, Markov processes with a discrete component. Chap. IV. Processes with independent increments: One- dimensional and multidimensional state space, first passage times, extremal values. Chap. V Markov branching processes: Finite number of particle types, continuous mass distribution, general Markov processes with branching. The book is closed by a bibliography and a short section with historical remarks.

This second volume presents a rather complete survey of the (now) classical theory of Markov processes following the treatment of such processes by Dynkin: Starting with measure theoretic definitions of Markov processes the associated operators are investigated. Excluded is the field of diffusion processes. The book is written on a mathematical rigorous level, presenting the theory without its applications. So a reader should have knowledge of the applications of Markov processes, and should be familiar with at least a part of the theory of stochastic processes. More or less it seems that the book is written for those who are specialists in the theory of Markov processes, or for those who want to become a specialist.

Reviewer: H.Daduna

##### MSC:

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

60Gxx | Stochastic processes |

60Jxx | Markov processes |

60K15 | Markov renewal processes, semi-Markov processes |