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A First Course in 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
A First Course in Stochastic Calculus
Softcover ISBN:  978-1-4704-6488-2
Product Code:  AMSTEXT/53
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6778-4
Product Code:  AMSTEXT/53.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-6488-2
eBook: ISBN:  978-1-4704-6778-4
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
A First Course in Stochastic Calculus
Click above image for expanded view
A First Course in 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
Softcover ISBN:  978-1-4704-6488-2
Product Code:  AMSTEXT/53
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6778-4
Product Code:  AMSTEXT/53.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-6488-2
eBook ISBN:  978-1-4704-6778-4
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 Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 532022; 270 pp
    MSC: 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.

    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.

    Ancillaries:

    Readership

    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.

  • 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 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
  • 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 well-written, 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, Feynman-Kac 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 so-called 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 measure-theoretic background. In short, it is a book of stochastic calculus applied to finance, relatively standard, but very up-to-date and with many interesting details. The numerical and computational approach is undoubtedly the most original aspect of the book. A strongly recommend-able book!

      Josep Vives, University of Barcelona
    • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 532022; 270 pp
MSC: 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.

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.

Ancillaries:

Readership

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 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
  • 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 well-written, 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, Feynman-Kac 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 so-called 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 measure-theoretic background. In short, it is a book of stochastic calculus applied to finance, relatively standard, but very up-to-date and with many interesting details. The numerical and computational approach is undoubtedly the most original aspect of the book. A strongly recommend-able book!

    Josep Vives, University of Barcelona
  • 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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