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.