ISBN: 3-540-54062-8
TITLE: Numerical Solution of Stochastic Differential Equations
AUTHOR: Kloeden, Peter E.; Platen, Eckhard
TOC:

Suggestions for the Reader xvii 
Basic Notation xxi 
Brief Survey of Stochastic Numerical Methods xxiii 
Part I. Preliminaries 
Chapter 1. Probability and Statistics 1 
1.1 Probabilities and Events 1 
1.2 Random Variables and Distributions 5 
1.3 Random Number Generators 11 
1.4 Moments 14 
1.5 Convergence of Random Sequences 22 
1.6 Basic Ideas About Stochastic Processes 26 
1.7 Diffusion Processes 34 
1.8 Wiener Processes and White Noise 40 
1.9 Statistical Tests and Estimation 44 
Chapter 2. Probability and Stochastic Processes 51 
2.1 Aspects of Measure and Probability Theory 51 
2.2 Integration and Expectations 55 
2.3 Stochastic Processes 63 
2.4 Diffusion and Wiener Processes 68 
Part II. Stochastic Differential Equations 
Chapter 3. Ito Stochastic Calculus 75 
3.1 Introduction 75 
3.2 The Ito Stochastic Integral 81 
3.3 The Ito Formula 90 
3.4 Vector Valued Ito Integrals 96 
3.5 Other Stochastic Integrals 99 
Chapter 4. Stochastic Differential Equations 103 
4.1 Introduction 103 
4.2 Linear Stochastic Differential Equations 110 
4.3 Reducible Stochastic Differential Equations 113 
4.4 Some Explicitly Solvable Equations 117 
4.5 The Existence and Uniqueness of Strong Solutions 127 
4.6 Strong Solutions as Diffusion Processes 141 
4.7 Diffusion Processes as Weak Solutions 144 
4.8 Vector Stochastic Differential Equations 148 
4.9 Stratonovich Stochastic Differential Equations 154 
Chapter 5. Stochastic Taylor Expansions 161 
5.1 Introduction 161 
5.2 Multiple Stochastic Integrals 167 
5.3 Coefficient Functions 177 
5.4 Hierarchical and Remainder Sets 180 
5.5 Ito-Taylor Expansions 181 
5.6 Stratonovich-Taylor Expansions 187 
5.7 Moments of Multiple Ito Integrals 190 
5.8 Strong Approximation of Multiple Stochastic Integrals 198 
5.9 Strong Convergence of Truncated Ito-Taylor Expansions 206 
5.10 Strong Convergence 
of Truncated Stratonovich-Taylor Expansions 210 
5.11 Weak Convergence of Truncated Ito-Taylor Expansions 211 
5.12 Weak Approximations of Multiple Ito Integrals 221 
Part III. Applications of Stochastic Differential Equations 
Chapter 6. Modelling with Stochastic 
Differential Equations 227 
6.1 Ito Versus Stratonovich 227 
6.2 Diffusion Limits of Markov Chains 229 
6.3 Stochastic Stability 232 
6.4 Parametric Estimation 241 
6.5 Optimal Stochastic Control 244 
6.6 Filtering 248 
Chapter 7. Applications of Stochastic Differential Equations 253 
7.1 Population Dynamics, Protein Kinetics and Genetics 253 
7.2 Experimental Psychology and Neuronal Activity 256 
7.3 Investment Finance and Option Pricing 257 
7.4 Turbulent Diffusion and Radio-Astronomy 259 
7.5 Helicopter Rotor and Satellite Orbit Stability 261 
7.6 Biological Waste Treatment, Hydrology and Air Quality 263 
7.7 Seismology and Structural Mechanics 266 
7.8 Fatigue Cracking, Optical Bistability 
and Nemantic Liquid Crystals 269 
7.9 Blood Clotting Dynamics and Cellular Energetics 271 
7.10 Josephson Tunneling Junctions Communications 
and Stochastic Annealing 273 
Part IV. Time Discrete Approximations 
Chapter 8. Time Discrete Approximation 
of Deterministic Differential Equations 277 
8.1 Introduction 277 
8.2 Taylor Approximations and Higher Order Methods 286 
8.3 Consistency, Convergence and Stability 292 
8.4 Roundoff Error 301 
Chapter 9. Introduction to Stochastic 
Time Discrete Approximation 305 
9.1 The Euler Approximation 305 
9.2 Example of a Time Discrete Simulation 307 
9.3 Pathwise Approximations 311 
9.4 Approximation of Moments 316 
9.5 General Time Discretizations and Approximations 321 
9.6 Strong Convergence and Consistency 323 
9.7 Weak Convergence and Consistency 326 
9.8 Numerical Stability 331 
Part V. Strong Approximations 
Chapter 10. Strong Taylor Approximations 339 
10.1 Introduction 339 
10.2 The Euler Scheme 340 
10.3 The Milstein Scheme 345 
10.4 The Order 1.5 Strong Taylor Scheme 351 
10.5 The Order 2.0 Strong Taylor Scheme 356 
10.6 General Strong Ito-Taylor Approximations 360 
10.7 General Strong Stratonovich-Taylor Approximations 365 
10.8 A Lemma on Multiple Ito Integrals 369 
Chapter 11. Explicit Strong Approximations 373 
11.1 Explicit Order 1.0 Strong Schemes 373 
11.2 Explicit Order 1.5 Strong Schemes 378 
11.3 Explicit Order 2.0 Strong Schemes 383 
11.4 Multistep Schemes 385 
11.5 General Strong Schemes 390 
Chapter 12. Implicit Strong Approximations 395 
12.1 Introduction 395 
12.2 Implicit Strong Taylor Approximations 396 
12.3 Implicit Strong Runge-Kutta Approximations 406 
12.4 Implicit Two-Step Strong Approximations 411 
12.5 A-Stability of Strong One-Step Schemes 417 
12.6 Convergence Proofs 420 
Chapter 13. Selected Applications 
of Strong Approximations 427 
13.1 Direct Simulation of Trajectories 427 
13.2 Testing Parametric Estimators 435 
13.3 Discrete Approximations for Markov Chain Filters 442 
13.4 Asymptotically Efficient Schemes 453 
Part VI. Weak Approximations 
Chapter 14. Weak Taylor Approximations 457 
14.1 The Euler Scheme 457 
14.2 The Order 2.0 Weak Taylor Scheme 464 
14.3 The Order 3.0 Weak Taylor Scheme 468 
14.4 The Order 4.0 Weak Taylor Scheme 470 
14.5 General Weak Taylor Approximations 472 
14.6 Leading Error Coefficients 480 
Chapter 15. Explicit and Implicit Weak Approximations 485 
15.1 Explicit Order 2.0 Weak Schemes 485 
15.2 Explicit Order 3.0 Weak Schemes 488 
15.3 Extrapolation Methods 491 
15.4 Implicit Weak Approximations 495 
15.5 Predictor-Corrector Methods 501 
15.6 Convergence of Weak Schemes 506 
Chapter 16. Variance Reduction Methods 511 
16.1 Introduction 511 
16.2 The Measure Transformation Method 513 
16.3 Variance Reduced Estimators 516 
16.4 Unbiased Estimators 522 
Chapter 17. Selected Applications of Weak Approximations 529 
17.1 Evaluation of Functional Integrals 529 
17.2 Approximation of Invariant Measures 540 
17.3 Approximation of Lyapunov Exponents 545 
Solutions of Exercises 549 
Bibliographical Notes 587 
Bibliography 599 
Index 629 
END