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Handbook of Brownian motion - facts and formulae. (English) Zbl 0859.60001

Probability and Its Applications. Basel: Birkhäuser. xiv, 462 p. (1996).
The monograph consists of two parts: theory and tables. The theoretical part provides the reader with a wealth of relevant facts on one-dimensional diffusions without proofs but with precise references. There are separate chapters on linear diffusions (including local time, additive functionals and an ergodic theorem), stochastic calculus with respect to Brownian motion, Brownian motion (including excursions) and sections on Ray-Knight theorems, Feynman-Kac formula etc. The larger Part II contains seven sections of tables related to Brownian motion, Brownian motion with drift, reflecting Brownian motion, Bessel processes of order zero, Bessel process of order \(1/2\), Bessel process of order \(\nu>0\) and the Ornstein-Uhlenbeck process. Each section is subdivided into four subsections containing tables of distributions (or their two Laplace transforms or characteristic functions) pertaining to (independent) exponential stopping, stopping at first hitting time, stopping at first exit time (from an interval) and stopping at inverse local time.
The authors have devised a systematic numbering of the formulae which greatly helps the reader to find the formula he is looking for. The book certainly fills a gap in the literature. Even though all the formulae are (essentially) known this is the first time that they have been collected and listed in a systematic manner. The monograph is likely to be useful for both mathematicians and scientists from other fields who use diffusions in applications.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60Jxx Markov processes
60J65 Brownian motion
60Gxx Stochastic processes
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