A First Course in Stochastic Calculus is a complete guide for advanced undergraduate students to take the next step in exploring probability theory and for master's students in mathematical finance who would like to build an intuitive and theoretical understanding of stochastic processes. This book is also an essential tool for finance professionals who wish to sharpen their knowledge and intuition about stochastic calculus.
Louis-Pierre Arguin: Baruch College, City University of New York, New York, NY and Graduate Center, City University of New York, New York, NY
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