Softcover ISBN:  9781470464882 
Product Code:  AMSTEXT/53 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470467784 
Product Code:  AMSTEXT/53.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470464882 
eBook: ISBN:  9781470467784 
Product Code:  AMSTEXT/53.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 
Softcover ISBN:  9781470464882 
Product Code:  AMSTEXT/53 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470467784 
Product Code:  AMSTEXT/53.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470464882 
eBook ISBN:  9781470467784 
Product Code:  AMSTEXT/53.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 

Book DetailsPure and Applied Undergraduate TextsVolume: 53; 2022; 270 ppMSC: Primary 60; 91
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.
LouisPierre 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 selfcontained account of stochastic calculus applications to option pricing in finance.
Ancillaries:
ReadershipUndergraduate and graduate students interested in advanced probability and the applications of stochastic calculus to finance. Finance professionals who want to develop their knowledge and intuition of stochastic calculus.

Table of Contents

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 FeynmanKac 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 CameronMartin Theorem

9.3. Extensions of the CameronMartin 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 BlackScholes Model

10.5. The Greeks

10.6. RiskNeutral 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


Additional Material

Reviews

Congratulations to both the author for writing this valuable book and the AMS for its publication as a volume in the prestigious series "Pure and Applied Undergraduate Texts." There are all good reasons to strongly recommended the book to the thousands of students worldwide studying stochastic calculus, in particular to students following MSc and PhD programs in the area of 'mathematical finance.' Teachers of courses in stochastic calculus can efficiently combine this book with other sources.
Jordan M. Stoyanov (Sofia), zbMathOpen 
The book is quite concise and very wellwritten, with many illustrative figures. A nice detail is that in almost all chapters, the topic is taken a little further than usual: gambler's ruin, Tanaka formula, Dirichlet problem, martingale representation, FeynmanKac formula, and Heston model! But more than this, the book has several nice and original details. Many exercises allow the reader to delve deeper into topics beyond the basic points, hinting at the paths to deeper levels of understanding of the theory. The historical notes are certainly interesting. And above all, the socalled numerical projects, proposed at every chapter, make the book clearly numerical and computational aspects oriented, a fact that is very important in the training of future quantitative analysts, and useful to progress into stochastic calculus without a significant measuretheoretic background. In short, it is a book of stochastic calculus applied to finance, relatively standard, but very uptodate and with many interesting details. The numerical and computational approach is undoubtedly the most original aspect of the book. A strongly recommendable book!
Josep Vives, University of Barcelona 
LouisPierre 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 prizewinning 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. LouisPierre Arguin's excellent and artfully designed text will give me the ideal vehicle to do so.
Ioannis Karatzas, Columbia University, New York


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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.
LouisPierre 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 selfcontained account of stochastic calculus applications to option pricing in finance.
Ancillaries:
Undergraduate and graduate students interested in advanced probability and the applications of stochastic calculus to finance. Finance professionals who want to develop their knowledge and intuition of stochastic calculus.

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 FeynmanKac 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 CameronMartin Theorem

9.3. Extensions of the CameronMartin 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 BlackScholes Model

10.5. The Greeks

10.6. RiskNeutral 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

Congratulations to both the author for writing this valuable book and the AMS for its publication as a volume in the prestigious series "Pure and Applied Undergraduate Texts." There are all good reasons to strongly recommended the book to the thousands of students worldwide studying stochastic calculus, in particular to students following MSc and PhD programs in the area of 'mathematical finance.' Teachers of courses in stochastic calculus can efficiently combine this book with other sources.
Jordan M. Stoyanov (Sofia), zbMathOpen 
The book is quite concise and very wellwritten, with many illustrative figures. A nice detail is that in almost all chapters, the topic is taken a little further than usual: gambler's ruin, Tanaka formula, Dirichlet problem, martingale representation, FeynmanKac formula, and Heston model! But more than this, the book has several nice and original details. Many exercises allow the reader to delve deeper into topics beyond the basic points, hinting at the paths to deeper levels of understanding of the theory. The historical notes are certainly interesting. And above all, the socalled numerical projects, proposed at every chapter, make the book clearly numerical and computational aspects oriented, a fact that is very important in the training of future quantitative analysts, and useful to progress into stochastic calculus without a significant measuretheoretic background. In short, it is a book of stochastic calculus applied to finance, relatively standard, but very uptodate and with many interesting details. The numerical and computational approach is undoubtedly the most original aspect of the book. A strongly recommendable book!
Josep Vives, University of Barcelona 
LouisPierre 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 prizewinning 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. LouisPierre Arguin's excellent and artfully designed text will give me the ideal vehicle to do so.
Ioannis Karatzas, Columbia University, New York