Brownian Motion, Martingales, and Stochastic Calculus

Abstract

This book originates from lecture notes for an introductory course on stochastic calculus taught as part of the master’s program in probability and statistics at Université Pierre et Marie Curie and then at Université Paris-Sud. The aim of this course was to provide a concise but rigorous introduction to the theory of stochastic calculus for continuous semimartingales, putting a special emphasis on Brownian motion. This book is intended for students who already have a good knowledge of advanced probability theory, including tools of measure theory and the basic properties of conditional expectation. We also assume some familiarity with the notion of uniform integrability (see, for instance, Chapter VII in Grimmett and Stirzaker [30]). For the reader’s convenience, we record in Appendix A2 those results concerning discrete time martingales that we use in our study of continuous time martingales. The first chapter is a brief presentation of Gaussian vectors and processes. The main goal is to arrive at the notion of a Gaussian white noise, which allows us to give a simple construction of Brownian motion in Chap. 2. In this chapter, we discuss the basic properties of sample paths of Brownian motion and the strong Markov propertywith its classical application to the reflection principle. Chapter 2 also gives us the opportunity to introduce, in the relatively simple setting of Brownian motion, the important notions of filtrations and stopping times, which are studied in a more systematic and abstract way in Chap. 3. The latter chapter discusses continuous time martingales and supermartingales and investigates the regularity properties of their sample paths. Special attention is given to the optional stopping theorem, which in connection with stochastic calculus yields a powerful tool for lots of explicit calculations. Chapter 4 introduces continuous semimartingales, starting with a detailed discussion of finite variation functions and processes.We then discuss local martingales, but as in most of the remaining part of the course, we restrict our attention to the case of continuous sample paths. We provide a detailed proof of the key theorem on the existence of the quadratic variation of a local martingale. Chapter 5 is at the core of this book, with the construction of the stochastic integral with respect to a continuous semimartingale, the proof in this setting of the celebrated Itô formula, and several important applications (Lévy’s characterization theorem for Brownian motion, the Dambis–Dubins–Schwarz representation of a continuous martingale as a time-changed Brownian motion, the Burkholder– Davis–Gundy inequalities, the representation of Brownian martingales as stochastic integrals, Girsanov’s theorem and the Cameron–Martin formula, etc.). Chapter 6, which presents the fundamental ideas of the theory of Markov processes with emphasis on the case of Feller semigroups, may appear as a digression to our main topic. The results of this chapter, however, play an important role in Chap. 7, where we combine tools of the theory of Markov processes with techniques of stochastic calculus to investigate connections of Brownian motion with partial differential equations, including the probabilistic solution of the classical Dirichlet problem. Chapter 7 also derives the conformal invariance of planar Brownian motion and applies this property to the skew-product decomposition, which in turn leads to asymptotic laws such as the celebrated Spitzer theorem on Brownian windings. Stochastic differential equations, which are another very important application of stochastic calculus and in fact motivated Itô’s invention of this theory, are studied in detail in Chap. 8, in the case of Lipschitz continuous coefficients. Here again the general theory developed in Chap. 6 is used in our study of the Markovian properties of solutions of stochastic differential equations, which play a crucial role in many applications. Finally, Chap. 9 is devoted to local times of continuous semimartingales. The construction of local times in this setting, the study of their regularity properties, and the proof of the density of occupation formula are very convincing illustrations of the power of stochastic calculus techniques.We conclude Chap. 9 with a brief discussion of Brownian local times, including Trotter’s theorem and the famous Lévy theorem identifying the law of the local time process at level 0.