An introduction to stochastic modeling. 3rd ed.

*(English)*Zbl 0946.60002
Boston, MA: Academic Press. 656 p. (1998).

For an earlier review of this monograph see Zbl 0796.60001.

The objective of this textbook is to introduce, on the basis of a solid course on elementary probability calculus, some theory and application of stochastic processes. The approach is elementary (no measure theory or general integral calculus is used) and, hence, it bridges the gap between basic probability know-how and an intermediate level course in stochastic processes. The reader is introduced to the standard concepts and methods of stochastic modelling.

In the Introduction (Chapter I) some basic results from elementary probability theory are given (discrete and continuous distributions, moments etc.). Chapter II deals with conditional probabilities, conditional expectations, random sums and martingales. Markov chains in discrete time are the topic of the Chapters III and IV. Poisson processes are considered in Chapter V. Chapter VI is devoted to continuous time Markov chains. Renewal processes and variations of them can be found in Chapter VII. The Brownian motion process and several applications of it and its variants in financial modelling (Black-Scholes formula) are introduced in Chapter VIII. Queueing models are the topic of Chapter IX.

The applicability of stochastic processes in various fields of sciences is widely illustrated by the many stochastic models and examples, it is worth noticing that, in addition to standard material and examples, the book also presents a lot of nonstandard models, like spatial Poisson processes, compound and marked Poisson processes, set-valued Markov processes. Furthermore, the book contains more than 250 exercises with answers, and 350 problems that are more difficult to solve; some of them involve extensive algebraic or calculus manipulations.

The objective of this textbook is to introduce, on the basis of a solid course on elementary probability calculus, some theory and application of stochastic processes. The approach is elementary (no measure theory or general integral calculus is used) and, hence, it bridges the gap between basic probability know-how and an intermediate level course in stochastic processes. The reader is introduced to the standard concepts and methods of stochastic modelling.

In the Introduction (Chapter I) some basic results from elementary probability theory are given (discrete and continuous distributions, moments etc.). Chapter II deals with conditional probabilities, conditional expectations, random sums and martingales. Markov chains in discrete time are the topic of the Chapters III and IV. Poisson processes are considered in Chapter V. Chapter VI is devoted to continuous time Markov chains. Renewal processes and variations of them can be found in Chapter VII. The Brownian motion process and several applications of it and its variants in financial modelling (Black-Scholes formula) are introduced in Chapter VIII. Queueing models are the topic of Chapter IX.

The applicability of stochastic processes in various fields of sciences is widely illustrated by the many stochastic models and examples, it is worth noticing that, in addition to standard material and examples, the book also presents a lot of nonstandard models, like spatial Poisson processes, compound and marked Poisson processes, set-valued Markov processes. Furthermore, the book contains more than 250 exercises with answers, and 350 problems that are more difficult to solve; some of them involve extensive algebraic or calculus manipulations.

Reviewer: A.Brandt (Berlin)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60Kxx | Special processes |

60Jxx | Markov processes |