Hjem

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.