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## Book Description

An introduction to the mathematical theory and financial models developed and used on Wall Street

Providing both a theoretical and practical approach to the underlying mathematical theory behind financial models, Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach presents important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus. Measure theory is indispensable to the rigorous development of probability theory and is also necessary to properly address martingale measures, the change of numeraire theory, and LIBOR market models. In addition, probability theory is presented to facilitate the development of stochastic processes, including martingales and Brownian motions, while stochastic processes and stochastic calculus are discussed to model asset prices and develop derivative pricing models.

The authors promote a problem-solving approach when applying mathematics in real-world situations, and readers are encouraged to address theorems and problems with mathematical rigor. In addition, Measure, Probability, and Mathematical Finance features:

• A comprehensive list of concepts and theorems from measure theory, probability theory, stochastic processes, and stochastic calculus

• Over 500 problems with hints and select solutions to reinforce basic concepts and important theorems

• Classic derivative pricing models in mathematical finance that have been developed and published since the seminal work of Black and Scholes

• Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach is an ideal textbook for introductory quantitative courses in business, economics, and mathematical finance at the upper-undergraduate and graduate levels. The book is also a useful reference for readers who need to build their mathematical skills in order to better understand the mathematical theory of derivative pricing models.

1. Cover
2. Half Title page
3. Title page
5. Dedication
6. Preface
7. Financial Glossary
8. Part I: Measure Theory
1. Chapter 1: Sets and Sequences
2. Chapter 2: Measures
3. Chapter 3: Extension of Measures
4. Chapter 4: Lebesgue-Stieltjes Measures
5. Chapter 5: Measurable Functions
6. Chapter 6: Lebesgue Integration
7. Chapter 7: The Radon-Nikodym Theorem
8. Chapter 8: LP Spaces
9. Chapter 9: Convergence
10. Chapter 10: Product Measures
9. Part II: Probability Theory
1. Chapter 11: Events and Random Variables
2. Chapter 12: Independence
3. Chapter 13: Expectation
4. Chapter 14: Conditional Expectation
5. Chapter 15: Inequalities
6. Chapter 16: Law of Large Numbers
7. Chapter 17: Characteristic Functions
8. Chapter 18: Discrete Distributions
9. Chapter 19: Continuous Distributions
10. Chapter 20: Central Limit Theorems
10. Part III: Stochastic Processes
1. Chapter 21: Stochastic Processes
2. Chapter 22: Martingales
3. Chapter 23: Stopping Times
4. Chapter 24: Martingale Inequalities
5. Chapter 25: Martingale Convergence Theorems
6. Chapter 26: Random Walks
7. Chapter 27: Poisson Processes
8. Chapter 28: Brownian Motion
9. Chapter 29: Markov Processes
10. Chapter 30: Lévy Processes
11. Part IV: Stochastic Calculus
1. Chapter 31: The Wiener Integral
2. Chapter 32: The Itô Integral
3. Chapter 33: Extension of the Itô Integral
4. Chapter 34: Martingale Stochastic Integrals
5. Chapter 35: The Itô Formula
6. Chapter 36: Martingale Representation Theorem
7. Chapter 37: Change of Measure
8. Chapter 38: Stochastic Differential Equations
9. Chapter 39: Diffusion
10. Chapter 40: The Feynman-Kac Formula
12. Part V: Stochastic Financial Models
1. Chapter 41: Discrete-Time Models
2. Chapter 42: Black-Scholes Option Pricing Models
3. Chapter 43: Path-Dependent Options
4. Chapter 44: American Options
5. Chapter 45: Short Rate Models
6. Chapter 46: Instantaneous Forward Rate Models
7. Chapter 47: Libor Market Models
13. References
14. List of Symbols
15. Subject Index