Description
Book SynopsisAn 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-w
Table of Contents
Preface xvii
Financial Glossary xxii
Part I Measure Theory
1 Sets and Sequences 3
2 Measures 15
3 Extension of Measures 29
4 Lebesgue-Stieltjes Measures 37
5 Measurable Functions 47
6 Lebesgue Integration 57
7 The Radon-Nikodym Theorem 77
8 LP Spaces 85
9 Convergence 97
10 Product Measures 113
Part II Probability Theory
11 Events and Random Variables 127
12 Independence 141
13 Expectation 161
14 Conditional Expectation 173
15 Inequalities 189
16 Law of Large Numbers 199
17 Characteristic Functions 217
18 Discrete Distributions 227
19 Continuous Distributions 239
20 Central Limit Theorems 257
Part III Stochastic Processes
21 Stochastic Processes 271
22 Martingales 291
23 Stopping Times 301
24 Martingale Inequalities 321
25 Martingale Convergence Theorems 333
26 Random Walks 343
27 Poisson Processes 357
28 Brownian Motion 373
29 Markov Processes 389
30 Lévy Processes 401
Part IV Stochastic Calculus
31 The Wiener Integral 421
32 The Itô Integral 431
33 Extension of the Itô Integral 453
34 Martingale Stochastic Integrals 463
35 The Itô Formula 477
36 Martingale Representation Theorem 495
37 Change of Measure 503
38 Stochastic Differential Equations 515
39 Diffusion 531
40 The Feynman-Kac Formula 547
Part V Stochastic Financial Models
41 Discrete-Time Models 561
42 Black-Scholes Option Pricing Models 579
43 Path-Dependent Options 593
44 American Options 609
45 Short Rate Models 629
46 Instantaneous Forward Rate Models 647
47 LIBOR Market Models 667
References 687
List of Symbols 703
Subject Index 707
