Handbook of Brownian Motion

There are two parts in this book. The first part is devoted mainly to the proper­ ties of linear diffusions in general and Brownian motion in particular. The second part consists of tables of distributions of functionals of Brownian motion and re­ lated processes. The primary aim of this book is to give an easy reference to a large number of facts and formulae associated to Brownian motion. We have tried to do this in a "handbook-style". By this we mean that results are given without proofs but are equipped with a reference where a proof or a derivation can be found. It is our belief and experience that such a material would be very much welcome by students and people working with applications of diffusions and Brownian motion. In discussions with many of our colleagues we have found that they share this point of view. Our original plan included more things than we were able to realize. It turned out very soon when trying to put the plan into practice that the material would be too wide to be published under one cover. Excursion theory, which most of the recent results concerning linear Brownian motion and diffusions can be classified as, is only touched upon slightly here, not to mention Brownian motion in several dimensions which enters only through the discussion of Bessel processes. On the other hand, much attention is given to the theory of local time.

Contenu

I: Theory.- I. Stochastic processes in general.- 1. Basic definitions.- 2. Markov processes, transition functions, resolvents, and generators.- 3. Feller processes, Feller-Dynkin processes, and the strong Markov property.- 4. Martingales.- II. Linear diffusions.- 1. Basic facts.- 2. Local time.- 3. Passage times.- 4. Additive functionals and killing.- 5. Excessive functions.- 6. Ergodic results.- III. Stochastic calculus.- 1. Stochastic integration with respect to Brownian motion.- 2. The Itô and Tanaka formulae.- 3. The Cameron-Martin-Girsanov transformation of measure.- 4. Stochastic differential equations - strong solutions.- 5. Stochastic differential equations - weak solutions.- 6. One-dimensional stochastic differential equations.- IV. Brownian motion.- 1. Definition and basic properties.- 2. Brownian local time.- 3. Excursions.- 4. Brownian bridge.- 5. Brownian motion with drift.- 6. Bessel processes.- V. Local time as a Markov process.- 1. Diffusion local time.- 2. Local time of Brownian motion.- 2. Local time of Brownian motion with drift.- 4. Local time of Bessel process.- VI. Differential systems associated to Brownian motion.- 1. The Feynman-Kac formula.- 2. Exponential stopping.- 3. Stopping at first exit time.- 4. Stopping at inverse additive functional.- Appendix 1. Briefly on some diffusions.- II: Tables of Distributions of Functionals of Brownian Motion and Related Processes.- 1. Brownian motion.- 1. Exponential stopping.- 2. Stopping at first hitting time.- 3. Stopping at first exit time.- 4. Stopping at inverse local time.- 2. Brownian motion with drift.- 1. Exponential stopping.- 2. Stopping at first hitting time.- 3. Stopping at first exit time.- 4. Stopping at inverse local time.- 3. Reflecting Brownian motion.- 1. Exponential stopping.- 2. Stopping at first hitting time.- 4. Stopping at inverse local time.- 4. Bessel process of order zero.- 1. Exponential stopping.- 2. Stopping at first hitting time.- 3. Stopping at first exit time.- 4. Stopping at inverse local time.- 5. Bessel process of order 1/2.- 1. Exponential stopping.- 2. Stopping at first hitting time.- 3. Stopping at first exit time.- 4. Stopping at inverse local time.- 6. Bessel process of order v > 0.- 1. Exponential stopping.- 2. Stopping at first hitting time.- 3. Stopping at first exit time.- 4. Stopping at inverse local time.- 7. Ornstein-Uhlenbeck process.- 1. Exponential stopping.- 2. Stopping at first hitting time.- 3. Stopping at first exit time.- 4. Stopping at inverse local time.- Appendix 2. Special functions.