Frontmatter Page vii Monday, March 8, 2004 10:06 AM
Contents viiClosed-Form Solutions of Ordinary Differential Equations 246
Linear Differential Equation 247
Numerical Solutions of Ordinary Differential Equations 249
The Finite Difference Method 249
Nonlinear Dynamics and Chaos 256
Fractals 258
Partial Differential Equations 259
Diffusion Equation 259
Solution of the Diffusion Equation 261
Numerical Solution of PDEs 263
Summary 265
CHAPTER 10
Stochastic Differential Equations 267
The Intuition Behind Stochastic Differential Equations 268
Itô Processes 271
The 1-Dimensional Itô Formula 272
Stochastic Differential Equations 274
Generalization to Several Dimensions 276
Solution of Stochastic Differential Equations 278
The Arithmetic Brownian Motion 280
The Ornstein-Uhlenbeck Process 280
The Geometric Brownian Motion 281
Summary 282
CHAPTER 11
Financial Econometrics: Time Series Concepts, Representations, and Models 283
Concepts of Time Series 284
Stylized Facts of Financial Time Series 286
Infinite Moving-Average and Autoregressive
Representation of Time Series 288
Univariate Stationary Series 288
The Lag Operator L 289
Stationary Univariate Moving Average 292
Multivariate Stationary Series 293
Nonstationary Series 295
ARMA Representations 297
Stationary Univariate ARMA Models 297
Nonstationary Univariate ARMA Models 300
Stationary Multivariate ARMA Models 301
Nonstationary Multivariate ARMA Models 304
Markov Coefficients and ARMA Models 304
Hankel Matrices and ARMA Models 305
State-Space Representation 305
Equivalence of State-Space and ARMA Representations 308
Integrated Series and Trends 309
Summary 313