Foreword
Preface

Chapter 1. Basic Notions of Probability
1.1. Probability Space
1.2. Random Variables and Their Distributions
1.3. Expectation
1.4. Inequalities
1.5. Numerical Projects and Exercices
Exercises
1.6. Historical and Bibliographical Notes

Chapter 2. Gaussian Processes
2.1. Random Vectors
2.2. Gaussian Vectors
2.3. Gaussian Processes
2.4. A Geometric Point of View
2.5. Numerical Projects and Exercises
Exercises
2.6. Historical and Bibliographical Notes

Chapter 3. Properties of Brownian Motion
3.1. Properties of the Distribution
3.2. Properties of the Paths
3.3. A Word on the Construction of Brownian Motion
3.4. A Point of Comparison: The Poisson Process
3.5. Numerical Projects and Exercises
Exercises
3.6. Historical and Bibliographical Notes

Chapter 4. Martingales
4.1. Elementary Conditional Expectation
4.2. Conditional Expectation as a Projection
4.3. Martingales
4.4. Computations with Martingales
4.5. Reflection Principle for Brownian Motion
4.6. Numerical Projects and Exercises
Exercises
4.7. Historical and Bibliographical Notes

Chapter 5. Itô Calculus
5.1. Preliminaries
5.2. Martingale Transform
5.3. The Itô Integral
5.4. Itô’s Formula
5.5. Gambler’s Ruin for Brownian Motion with Drift
5.6. Tanaka’s Formula
5.7. Numerical Projects and Exercises
Exercises
5.8. Historical and Bibliographical Notes

Chapter 6. Multivariate Itô Calculus
6.1. Multidimensional Brownian Motion
6.2. Itô’s Formula
6.3. Recurrence and Transience of Brownian Motion
6.4. Dynkin’s Formula and the Dirichlet Problem
6.5. Numerical Projects and Exercises
Exercises
6.6. Historical and Bibliographical Notes

Chapter 7. Itô Processes and Stochastic Differential Equations
7.1. Definition and Examples
7.2. Itô’s Formula
7.3. Multivariate Extension
7.4. Numerical Simulations of SDEs
7.5. Existence and Uniqueness of Solutions of SDEs
7.6. Martingale Representation and Lévy’s Characterization
7.7. Numerical Projects and Exercises
Exercises
7.8. Historical and Bibliographical Notes

Chapter 8. The Markov Property
8.1. The Markov Property for Diffusions
8.2. The Strong Markov Property
8.3. Kolmogorov’s Equations
8.4. The Feynman-Kac Formula
8.5. Numerical Projects and Exercises
Exercises
8.6. Historical and Bibliographical Notes
Chapter 9. Change of Probability
9.1. Change of Probability for a Random Variable
9.2. The Cameron-Martin Theorem
9.3. Extensions of the Cameron-Martin Theorem
9.4. Numerical Projects and Exercises
Exercises
9.5. Historical and Bibliographical Notes
Chapter 10. Applications to Mathematical Finance
10.1. Market Models
10.2. Derivatives
10.3. No Arbitrage and Replication
10.4. The Black-Scholes Model
10.5. The Greeks
10.6. Risk-Neutral Pricing
10.7. Exotic Options
10.8. Interest Rate Models
10.9. Stochastic Volatility Models
10.10. Numerical Projects and Exercises
Exercises
10.11. Historical and Bibliographical Notes
Bibliography