Frontmatter Page vi Monday, March 8, 2004 10:06 AM
vi ContentsOutcomes and Events 169
Probability 170
Measure 171
Random Variables 172
Integrals 172
Distributions and Distribution Functions 174
Random Vectors 175
Stochastic Processes 178
Probabilistic Representation of Financial Markets 180
Information Structures 181
Filtration 182
Conditional Probability and Conditional Expectation 184
Moments and Correlation 186
Copula Functions 188
Sequences of Random Variables 189
Independent and Identically Distributed Sequences 191
Sum of Variables 191
Gaussian Variables 194
The Regression Function 197
Linear Regression 197
Summary 199
CHAPTER 7
Optimization 201
Maxima and Minima 202
Lagrange Multipliers 204
Numerical Algorithms 206
Linear Programming 206
Quadratic Programming 211
Calculus of Variations and Optimal Control Theory 212
Stochastic Programming 214
Summary 216
CHAPTER 8
Stochastic Integrals 217
The Intuition Behind Stochastic Integrals 219
Brownian Motion Defined 225
Properties of Brownian Motion 230
Stochastic Integrals Defined 232
Some Properties of Itô Stochastic Integrals 236
Summary 237
CHAPTER 9
Differential Equations and Difference Equations 239
Differential Equations Defined 240
Ordinary Differential Equations 240
Order and Degree of an ODE 241
Solution to an ODE 241
Systems of Ordinary Differential Equations 243