1 Martingales Stopping Times and Filtrations. - 1. 1. Stochastic Processes and ?-Fields. - 1. 2. Stopping Times. - 1. 3. Continuous-Time Martingales. - 1. 4. The DoobMeyer 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. Language: English