Stochastic calculus. A practical introduction.

*(English)*Zbl 0856.60002
Probability and Stochastics Series. Boca Raton, FL: CRC Press. vi, 341 p. (1996).

This book is sold under headings such as “time-saving”, “compact”, “thorough” or “the intellectual equivalent of the ginzu knife”. The main question seems whether a book of 340 pages is compact or not; a brief summary of the contents will help to explain the other epithets.

Chapter 1 introduces Brownian motion: definition, construction and Markov properties. Chapter 2 explains in detail stochastic integration for continuous semimartingales; apart from the basic construction and properties, this includes Itô’s formula, local time via the Meyer-Tanaka formula and Girsanov’s theorem. Chapter 3 continues the discussion of Brownian motion: recurrence and transience, occupation and exit times, Burkholder-Davis-Gundy inequalities, Lévy’s characterization and representations of continuous local martingales as time-changes or stochastic integrals of Brownian motion. Chapter 4 treats the connections between Brownian motion and partial differential equations: probabilistic solutions of the Dirichlet problem and the heat, Poisson and Schrödinger equations, the Feynman-Kac formula and several applications of analytic results to formulas for Brownian motion. Chapter 5 discusses stochastic differential equations: strong solutions via Picard’s method, the Yamada-Watanabe refinement in one dimension, weak solutions and martingale problems, change of drift via Girsanov’s theorem and change of diffusion via time change. Chapter 6 contains a treatment of solutions of one-dimensional SDEs: Feller’s test for explosion, recurrence and transience, Green function, boundary behaviour and applications to higher dimensions. Chapter 7 is a short introduction to Markov processes; central definitions are followed by the two basic examples of pure jump processes and diffusions, and existence of and convergence to an invariant distribution are discussed via embedded Harris chains. Chapter 8 is an introduction to weak convergence: generalities in metric spaces, Prokhorov’s theorem, tightness in \(C\) and \(D\), Donsker’s theorem in \(C\), convergence of Markov chains to diffusions and a number of examples.

For anyone teaching a first course in stochastic calculus, this book will turn out to be time-saving by “bringing together in one place useful material that is scattered in a variety of sources”. Thanks to a lively style and good explanations, it should also be attractive for advanced students who can get a quick yet thorough overview of stochastic calculus before turning to other sources for more detailed accounts of special topics. Exercises with solutions are another useful feature, and so the overall impression is that of a most welcome addition to the existing literature.

P.S. A ginzu knife is a Japanese kitchen knife with an extremely sharp blade; it is remarkable for the multitude of tasks it can be used for.

Chapter 1 introduces Brownian motion: definition, construction and Markov properties. Chapter 2 explains in detail stochastic integration for continuous semimartingales; apart from the basic construction and properties, this includes Itô’s formula, local time via the Meyer-Tanaka formula and Girsanov’s theorem. Chapter 3 continues the discussion of Brownian motion: recurrence and transience, occupation and exit times, Burkholder-Davis-Gundy inequalities, Lévy’s characterization and representations of continuous local martingales as time-changes or stochastic integrals of Brownian motion. Chapter 4 treats the connections between Brownian motion and partial differential equations: probabilistic solutions of the Dirichlet problem and the heat, Poisson and Schrödinger equations, the Feynman-Kac formula and several applications of analytic results to formulas for Brownian motion. Chapter 5 discusses stochastic differential equations: strong solutions via Picard’s method, the Yamada-Watanabe refinement in one dimension, weak solutions and martingale problems, change of drift via Girsanov’s theorem and change of diffusion via time change. Chapter 6 contains a treatment of solutions of one-dimensional SDEs: Feller’s test for explosion, recurrence and transience, Green function, boundary behaviour and applications to higher dimensions. Chapter 7 is a short introduction to Markov processes; central definitions are followed by the two basic examples of pure jump processes and diffusions, and existence of and convergence to an invariant distribution are discussed via embedded Harris chains. Chapter 8 is an introduction to weak convergence: generalities in metric spaces, Prokhorov’s theorem, tightness in \(C\) and \(D\), Donsker’s theorem in \(C\), convergence of Markov chains to diffusions and a number of examples.

For anyone teaching a first course in stochastic calculus, this book will turn out to be time-saving by “bringing together in one place useful material that is scattered in a variety of sources”. Thanks to a lively style and good explanations, it should also be attractive for advanced students who can get a quick yet thorough overview of stochastic calculus before turning to other sources for more detailed accounts of special topics. Exercises with solutions are another useful feature, and so the overall impression is that of a most welcome addition to the existing literature.

P.S. A ginzu knife is a Japanese kitchen knife with an extremely sharp blade; it is remarkable for the multitude of tasks it can be used for.

Reviewer: M.Schweizer (Berlin)

##### MSC:

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

60Hxx | Stochastic analysis |