Contents
Reader Guidelines
1 Preliminaries
1.1 Basic Concepts from Probability Theory
1.1.1 Random Variables . . . . . . .
1.1.2 Random Vectors . . . . . . . .
1.1.3 Independence and Dependence
1.2 Stochastic Processes . . . .
1.3 Brownian Motion . . . . . . . . . . . .
1.3.1 Defining Properties. . . . . . .
1.3.2 Processes Derived from Brownian J\'Iotion
1.3.3 Simulation of Brownian Sample Paths . .
1.4 Conditional Expectation . . . . . . . . . . . . . .
1.4.1 Conditional Expectation under Discrete Condition
1.4.2 About a-Fields . . . . . . . . . . . . . . . . . . . .
1.4.3 The General Conditional Expectation . . . . . . .
1.4.4 Rules for the Calculation of Conditional Expectations
1.4.5 The Projection Property of Conditional Expectations
1.5 Martingales . . . . . . . . .
1.5.1 Defining Properties. . . . . . . . . . . . . . . . . .
1.5.2 Examples.......................
1.5.3 The Interpretation of a Martingale as a Fair Game
2 The Stochastic Integral
2.1 The Riemann and Riemarm-Stieltjes Integrals.
2.1.1 The Ordinary Riemann Integral.
2.1.2 The Riemann-Stieltjes Integral
2.2 The Ito Integral. . . . . . . . .
2.2.1 A Motivating Example. . . . . 

2.2.2 The Ito Stochastic Integral for Simple Processes
2.2.3 The General Ito Stochastic Integral. . . . .
2.3 The Ito Lemma . . . . . . . . . . . . . . . . . . . .
2.3.1 The Classical Chain Rule of Differentiation
2.3.2 A Simple Version of the Ito Lemma
2.3.3 Extended Versions of the Ito Lemma
2.4 The Stratonovich and Other Integrals
A3 Non-Differentiability and Unbounded Variation of Brownian Sam-
ple Paths .................... . . . . . . . ., 188
A4 Proof of the Existence of the General Ito Stochastic Integral.. 190
A5 The Radon-Nikodym Theorem . . . . . . . . . . . . . . . . .. 193
A6 Proof of the Existence and Uniqueness of the Conditional Ex-
pectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 194
Bibliography
195
3 Stochastic Differential Equations 131
3.1 Deterministic Differential Equations . . . . . . . . 132
3.2 Ito Stochastic Differential Equations . . . . . . . . 134
3.2.1 What is a Stochastic Differential Equation? 134
3.2.2 Solving Ito Stochastic Differential Equations by the Ito
Lemma .. . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.2.3 Solving Ito Differential Equations via Stratonovich Cal-
culus . . . . . . . . . . . . . . . . . . . 145
3.3 The General Linear Differential Equation ......... 150
3.3.1 Linear Equations with Additive Noise . . . . . . . 150
3.3.2 Homogeneous Equations with Multiplicative Noise 153
3.3.3 The General Case .................. 155
3.3.4 The Expectation and Variance Functions of the Solution 156
3.4 Numerical Solution . . . . . . . . . . 157
3.4.1 The Euler Approximation . . 158
3.4.2 The Milstein Approximation 162
Index
199
List of Abbreviations and Symbols
209
4 Applications of Stochastic Calculus in Finance 167
4.1 The Black-Scholes Option Pricing Formula 168
4.1.1 A Short Excursion into Finance. . . . . 168
4.1.2 What is an Option? .. . . . . . . . . . 170
4.1.3 A Mathematical Formulation of the Option Pricing Pro-
blem . . . . .'. . . . . . . . . . . 172
4.1.4 The Black and Scholes Formula. . . . . . . . . 174
4.2 A Useful Technique: Change of Measure. . . . . . . . 176
4.2.1 What is a Change of the Underlying Measure? 176
4.2.2 An Interpretation of the Black-Scholes Formula by Chan-
ge of Measure. . . . . . . . . . . . . . . . . . . . . . " 180
Appendix 185
Al Modes of Convergence 185
A2 Inequalities . . . . . . 187