Brownian Motion and Stochastic Calculus

This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization).

This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises.

1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- 1.4. The Doob—Meyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- 2.5. The Markov Property.- 2.6. The Strong Markov Property and the Reflection Principle.- 2.7. Brownian Filtrations.- 2.8. Computations Based on Passage Times.- 2.9. The Brownian Sample Paths.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- 3.3. The Change-of-Variable Formula.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- 3.5. The Girsanov Theorem.- 3.6. Local Time and a Generalized Itô Rule for Brownian Motion.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- 4.3. The One-Dimensional Heat Equation.- 4.4. The Formulas of Feynman and Kac.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- 5.3. Weak Solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- 5.5. A Study of the One-Dimensional Case.- 5.6. Linear Equations.- 5.7. Connections with Partial Differential Equations.- 5.8. Applications to Economics.- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Lévy’s Theory of Brownian Local Time.-6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- 6.3. Two Independent Reflected Brownian Motions.- 6.4. Elastic Brownian Motion.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.