A First Course in Stochastic Calculus

Description

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 offers an exceptionally clear introduction to Brownian motion and to random processes governed by the principles of stochastic calculus. The beauty and power of the subject are made accessible to readers with a basic knowledge of probability, linear algebra, and multivariable calculus. This is achieved by emphasizing numerical experiments using elementary Python coding to build intuition and adhering to a rigorous geometric point of view on the space of random variables. This unique approach is used to elucidate the properties of Gaussian processes, martingales, and diffusions. One of the book's highlights is a detailed and self-contained account of stochastic calculus applications to option pricing in finance. Louis-Pierre Arguin's masterly introduction to stochastic calculus seduces the reader with its quietly conversational style; even rigorous proofs seem natural and easy. Full of insights and intuition, reinforced with many examples, numerical projects, and exercises, this book by a prize-winning mathematician and great teacher fully lives up to the author's reputation. I give it my strongest possible recommendation. ―Jim Gatheral, Baruch College I happen to be of a different persuasion, about how stochastic processes should be taught to undergraduate and MA students. But I have long been thinking to go against my own grain at some point and try to teach the subject at this level―together with its applications to finance―in one semester. Louis-Pierre Arguin's excellent and artfully designed text will give me the ideal vehicle to do so. ―Ioannis Karatzas, Columbia University, New York Contents 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 Index