Continuous Martingales and Brownian Motion

Frontmatter

Chapter 0. Preliminaries

Abstract

In this chapter, we review a few basic facts, mainly from integration and classical probability theories, which will be used throughout the book without further ado. Some other prerequisites, usually from calculus, which will be used in some special parts are collected in the Appendix at the end of the book.

Daniel Revuz, Marc Yor

Chapter I. Introduction

Abstract

A stochastic process is a phenomenon which evolves in time in a random way. Nature, everyday life, science offer us a huge variety of such phenomena or at least of phenomena which can be thought of as a function both of time and of a random factor. Such are for instance the price of certain commodities, the size of some populations, or the number of particles registered by a Geiger counter.

Daniel Revuz, Marc Yor

Chapter II. Martingales

Abstract

Martingales are a very important subject in their own right as well as by their relationship with analysis. Their kinship to BM will make them one of our main subjects of interest as well as one of our foremost tools. In this chapter, we describe some of their basic properties which we shall use throughout the book.

Daniel Revuz, Marc Yor

Chapter III. Markov Processes

Abstract

This chapter contains an introduction to Markov processes. Its relevance to our discussion stems from the fact that Brownian motion, as well as many processes which arise naturally in its study, are Markov processes; they even have the strong Markov property which is used in many applications. This chapter is also the occasion to introduce the Brownian filtrations which will appear frequently in the sequel.

Daniel Revuz, Marc Yor

Chapter IV. Stochastic Integration

Abstract

In this chapter, we introduce some basic techniques and notions which will be used throughout the sequel. Once and for all, we consider below, a filtered probability space (Ω, F, F _t , P) and we suppose that each F _t contains all the sets of P-measure zero in F. As a result, any limit (almost-sure, in the mean, etc.) of adapted processes is an adapted process; a process which is indistinguishable from an adapted process is adapted.

Daniel Revuz, Marc Yor

Chapter V. Representation of Martingales

Abstract

In this chapter, we take up the study of Brownian motion and, more generally, of continuous martingales. We will use the stochastic integration of Chap. IV together with the technique of time changes to be introduced presently.

Daniel Revuz, Marc Yor

Chapter VI. Local Times

Abstract

With Itô’s formula, we saw how C ²-functions operate on continuous semimartingales. We now extend this to convex functions, thus introducing the important notion of local time.

Daniel Revuz, Marc Yor

Chapter VII. Generators and Time Reversal

Abstract

In this chapter, we take up the study of Markov processes. We assume that the reader has read Sect. 1 and 2 in Chap. III.

Daniel Revuz, Marc Yor

Chapter VIII. Girsanov’s Theorem and First Applications

Abstract

In this chapter we study the effect on the space of continuous semimartingales of an absolutely continuous change of probability measure. The results we describe have far-reaching consequences from the theoretical point of view as is hinted at in Sect. 2; they also permit many explicit computations as is seen in Sect. 3.

Daniel Revuz, Marc Yor

Chapter IX. Stochastic Differential Equations

Abstract

In previous chapters stochastic differential equations have been mentioned several times in an informal manner.

Daniel Revuz, Marc Yor

Chapter X. Additive Functionals of Brownian Motion

Abstract

Although we want as usual to focus on the case of linear BM, we shall for a while consider a general Markov process for which we use the notation and results of Chap. III.

Daniel Revuz, Marc Yor

Chapter XI. Bessel Processes and Ray-Knight Theorems

Abstract

In this section, we take up the study of Bessel processes which was begun in Sect. 3 of Chap. VI and we use the notation thereof. We first make the following remarks.

Daniel Revuz, Marc Yor

Chapter XII. Excursions

Abstract

Throughout this section, we consider a measurable space (U, U) to which is added a point δ and we set U _δ = U ∪ {δ},U _δ = σ (U, {δ}).

Daniel Revuz, Marc Yor

Chapter XIII. Limit Theorems in Distribution

Abstract

In this section, we will specialize the notions of Sect. 5 Chap. 0 to the Wiener space W ^d . This space is a Polish space when endowed with the topology of uniform convergence on compact subsets of ℝ₊.

Daniel Revuz, Marc Yor

Backmatter