Numerical Solution of Stochastic Differential Equations
by Peter E. Kloeden and Eckhard Platen

Part I. Preliminaries

Chapter 1. Probability and Statistics

1. Probabilities and Events
2. Random Variables and Distributions
3. Random Number Generators
4. Moments
5. Convergence of Random Sequences
6. Basic Ideas About Stochastic Processes
7. Diffusion Processes
8. Wiener Processes and White Noise
9. Statistical Tests and Estimation

Chapter 2. Probability and Stochastic Processes

1. Aspects of Measure and Probability Theory
2. Integration and Expectation
3. Stochastic Processes
4. Diffusion and Wiener Processes

Part II. Stochastic Differential Equations

Chapter 3. Ito Stochastic Calculus

1. Introduction
2. The Ito Stochastic Integral
3. The Ito Formula
4. Vector Valued Ito Integrals
5. Other Stochastic Integrals

Chapter 4. Stochastic Differential Equations

1. Introduction
2. Linear Stochastic Differential Equations
3. Reducible Stochastic Differential Equations
4. Some Explicitly Solvable Equations
5. The Existence and Uniqueness of Strong Solutions
6. Strong Solutions as Diffusion Processes
7. Diffusion Processes as Weak Solutions
8. Vector Stochastic Differential Equations
9. Stratonovich Stochastic Differential Equations

Chapter 5. Stochastic Taylor Expansions

1. Introduction
2. Multiple Stochastic Integrals
3. Coefficient Functions
4. Hierarchical and Remainder Sets
5. Ito-Taylor Expansions
6. Stratonovich-Taylor Expansions
7. Moments of Multiple Ito Integrals
8. Strong Approximation of Multiple Stochastic Integrals
9. Strong Convergence of Truncated Ito-Taylor Expansions
10. Strong Convergence of Truncated Stratonovich-Taylor Expansions
11. Weak Convergence of Truncated Ito-Taylor Expansions
12. Weak Approximation of Multiple Ito Integrals

Part III. Applications of Stochastic Differential Equations

Chapter 6. Modelling with Stochastic Differential Equations

1. Ito Versus Stratonovich
2. Diffusion Limits of Markov Chains
3. Stochastic Stability
4. Parametric Estimation
5. Optimal Stochastic Control
6. Filtering

Chapter 7. Applications of Stochastic Differential Equations

1. Population Dynamics, Protein Kinetics and Genetics
2. Experimental Psychology and Neuronal Activity
3. Investment Finance and Option Pricing
4. Turbulent Diffusion and Radio-Astronomy
5. Helicopter Rotor and Satellite Orbit Stability
6. Biological Waste Treatment, Hydrology and Air Quality
7. Seismology and Structural Mechanics
8. Fatigue Cracking, Optical Bistability and Nemantic Liquid Crystals
9. Blood Clotting Dynamics and Cellular Energetics
10. Josephson Tunneling Junctions, Communications and Stochastic Annealing

Part IV. Time Discrete Approximations

Chapter 8. Time Discrete Approximation of Deterministic Differential Equations

1. Introduction
2. Taylor Approximations and Higher Order Methods
3. Consistency, Convergence and Stability
4. Roundoff Error

Chapter 9. Introduction to Stochastic Time Discrete Approximation

1. The Euler Approximation
2. Example of a Time Discrete Simulation
3. Pathwise Approximations
4. Approximation of Moments
5. General Time Discretizations and Approximations
6. Strong Convergence and Consistency
7. Weak Convergence and Consistency
8. Numerical Stability

Part V. Strong Approximations

Chapter 10. Strong Taylor Approximations

1. Introduction
2. The Euler Scheme
3. The Milstein Scheme
4. The Order 1.5 Strong Taylor Scheme
5. The Order 2.0 Strong Taylor Scheme
6. General Strong Ito-Taylor Approximations
7. General Strong Stratonovich-Taylor Approximations
8. A Lemma on Multiple Ito Integrals

Chapter 11. Explicit Strong Approximations

1. Explicit Order 1.0 Strong Schemes
2. Explicit Order 1.5 Strong Schemes
3. Explicit Order 2.0 Strong Schemes
4. Multistep Schemes
5. General Strong Schemes

Chapter 12. Implicit Strong Approximations

1. Introduction
2. Implicit Strong Taylor Approximations
3. Implicit Strong Runge-Kutta Approximations
4. Implicit Two-Step Strong Approximations
5. A-Stability of Strong One-Step Schemes
6. Convergence Proofs

Chapter 13. Selected Applications of Strong Approximations

1. Direct Simulation of Trajectories
2. Testing Parametric Estimators
3. Discrete Approximations for Markov Chain Filters
4. Asymptotically Efficient Schemes

Part VI. Weak Approximations

Chapter 14. Weak Taylor Approximations

1. The Euler Scheme
2. The Order 2.0 Weak Taylor Scheme
3. The Order 3.0 Weak Taylor Scheme
4. The Order 4.0 Weak Taylor Scheme
5. General Weak Taylor Approximations
6. Leading Error Coefficients

Chapter 15. Explicit and Implicit Weak Approximations

1. Explicit Order 2.0 Weak Schemes
2. Explicit Order 3.0 Weak Schemes
3. Extrapolation Methods
4. Implicit Weak Approximations
5. Predictor-Corrector Methods
6. Convergence of Weak Schemes

Chapter 16. Variance Reduction Methods

1. Introduction
2. The Measure Transformation Method
3. Variance Reduced Estimators
4. Unbiased Estimators

Chapter 17. Selected Applications of Weak Approximations

1. Evaluation of Functional Integrals
2. Approximation of Invariant Measures
3. Approximation of Lyapunov Exponents

	Numerical Solution of Stochastic Differential Equations
	by Peter E. Kloeden and Eckhard Platen