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Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

Book Description

Mathematical finance requires the use of advanced mathematical techniques drawn from the theory of probability, stochastic processes and stochastic differential equations. These areas are generally introduced and developed at an abstract level, making it problematic when applying these techniques to practical issues in finance.

Problems and Solutions in Mathematical Finance Volume I: Stochastic Calculus is the first of a four-volume set of books focusing on problems and solutions in mathematical finance.

This volume introduces the reader to the basic stochastic calculus concepts required for the study of this important subject, providing a large number of worked examples which enable the reader to build the necessary foundation for more practical orientated problems in the later volumes. Through this application and by working through the numerous examples, the reader will properly understand and appreciate the fundamentals that underpin mathematical finance.

Written mainly for students, industry practitioners and those involved in teaching in this field of study, Stochastic Calculus provides a valuable reference book to complement one's further understanding of mathematical finance.

Table of Contents

  1. Title Page
  2. Copyright
  3. Dedication
  4. Preface
  5. Prologue
    1. In The Beginning Was The Motion…
  6. About the Authors
  7. Chapter 1: General Probability Theory
    1. 1.1 Introduction
    2. 1.2 Problems and Solutions
  8. Chapter 2: Wiener Process
    1. 2.1 Introduction
    2. 2.2 Problems and Solutions
  9. Chapter 3: Stochastic Differential Equations
    1. 3.1 Introduction
    2. 3.2 Problems and Solutions
  10. Chapter 4: Change of Measure
    1. 4.1 Introduction
    2. 4.2 Problems and Solutions
  11. Chapter 5: Poisson Process
    1. 5.1 Introduction
    2. 5.2 Problems and Solutions
  12. Appendix A: Mathematics Formulae
    1. Indices
    2. Surds
    3. Exponential and Natural Logarithm
    4. Quadratic Equation
    5. Binomial Formula
    6. Series
    7. Summation
    8. Trigonometric Functions
    9. Hyperbolic Functions
    10. Complex Numbers
    11. Derivatives
    12. Standard Differentiations
    13. Taylor Series
    14. Maclaurin Series
    15. Landau Symbols and Asymptotics
    16. L'Hospital Rule
    17. Indefinite Integrals
    18. Standard Indefinite Integrals
    19. Definite Integrals
    20. Derivatives of Definite Integrals
    21. Integration by Parts
    22. Integration by Substitution
    23. Gamma Function
    24. Beta Function
    25. Convex Function
    26. Dirac Delta Function
    27. Heaviside Step Function
    28. Fubini's Theorem
  13. Appendix B: Probability Theory Formulae
    1. Probability Concepts
    2. Bayes' Rule
    3. Indicator Function
    4. Discrete Random Variables
    5. Univariate Case
    6. Bivariate Case
    7. Continuous Random Variables
    8. Univariate Case
    9. Bivariate Case
    10. Properties of Expectation and Variance
    11. Properties of Moment Generating and Characteristic Functions
    12. Correlation Coefficient
    13. Convolution
    14. Discrete Distributions
    15. Continuous Distributions
    16. Integrable and Square Integrable Random Variables
    17. Convergence of Random Variables
    18. Relationship Between Modes of Convergence
    19. Dominated Convergence Theorem
    20. Monotone Convergence Theorem
    21. The Weak Law of Large Numbers
    22. The Strong Law of Large Numbers
    23. The Central Limit Theorem
  14. Appendix C: Differential Equations Formulae
    1. Separable Equations
    2. First-Order Ordinary Differential Equations
    3. Second-Order Ordinary Differential Equations
    4. Homogeneous Heat Equations
    5. Stochastic Differential Equations
    6. Black–Scholes Model
    7. Black Model
    8. Garman–Kohlhagen Model
  15. Bibliography
  16. Notation
    1. Set Notation
    2. Mathematical Notation
    3. Probability Notation
  17. Index
  18. End User License Agreement