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Description of Numerical Solution of Stochastic Differential Equations |
The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations due to the peculiarities of stochastic calculus and stochastic differential equations, in both theory and applications, emphasizing the numerical methods needed to solve such equations.
It assumes of the reader an undergraduate background in mathematical methods typical of engineers and physicists, though many chapters begin with a descriptive summary, accessible to others who only require numerical recipes. To help the reader to develop an intuitive understanding of the underlying mathematics and hand-on numerical skills, exercises and over 100 PC-Exercises are included. The stochastic Taylor expansion provides discrete time numerical methods for stochastic differential equations.
The book presents many new results on higher-order methods for strong sample path approximations and for weak functional approximations, including implicit, predictor-corrector, extrapolation and variance-reduction methods. Besides serving as a basic text on such methods, the book offers the reader ready access to a large number of potential research problems in a field that is just beginning to expand rapidly and is widely applicable.
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Contents of Numerical Solution of Stochastic Differential Equations |
Suggestions for the Reader
Basic Notation
Brief Survey of Stochastic Numerical Methods
PART I. Preliminaries
1. Probability and Statistics
1.1 Probabilities and Events
1.2 Random Variables and Distributions
1.3 Random Number Generators
1.4 Moments
1.5 Convergence of Random Sequences
1.6 Basic Ideas About Stochastic Processes
1.7 Diffusion Processes
1.8 Wiener Processes and White Noise
1.9 Statistical Tests and Estimation
2. Probability and Stochastic Processes
2.1 Aspects of Measure and Probability Theory
2.2 Integration and Expectations
2.3 Stochastic Processes
2.4 Diffusion and Wiener Processes
PART II. Stochastic Differential Equations
3. Ito Stochastic Calculus
3.1 Introduction
3.2 The Ito Stochastic Integral
3.3 The Ito Formula
3.4 Vector Valued Ito Integrals
3.5 Other Stochastic Integrals
4. Stochastic Differential Equations
4.1 Introduction
4.2 Linear Stochastic Differential Equations
4.3 Reducible Stochastic Differential Equations
4.4 Some Explicitly Solvable Equations
4.5 The Existence and Uniqueness of Strong Solutions
4.6 Strong Solutions as Diffusion Processes
4.7 Diffusion Processes as Weak Solutions
4.8 Vector Stochastic Differential Equations
4.9 Stratonovich Stochastic Differential Equations
5. Stochastic Taylor Expansions
5.1 Introduction
5.2 Multiple Stochastic Integrals
5.3 Coefficient Functions
5.4 Hierarchical and Remainder Sets
5.5 Ito-Taylor Expansions
5.6 Stratonovich-Taylor Expansions
5.7 Moments of Multiple Ito Integrals
5.8 Strong Approximation of Multiple Stochastic Integrals
5.9 Strong Convergence of Truncated Ito-Taylor Expansions
5.10 Strong Convergence of Truncated Stratonovich-Taylor Expansions
5.11 Weak Convergence of Truncated Ito-Taylor Expansions
5.12 Weak Approximations of Multiple Ito Integrals
PART III. Applications of Stochastic Differential Equations
6. Modelling with Stochastic Differential Equations
6.1 Ito Versus Stratonovich
6.2 Diffusion Limits of Markov Chains
6.3 Stochastic Stability
6.4 Parametric Estimation
6.5 Optimal Stochastic Control
6.6 Filtering
7. Applications of Stochastic Differential Equations
7.1 Population Dynamics, Protein Kinetics and Genetics
7.2 Experimental Psychology and Neuronal Activity
7.3 Investment Finance and Option Pricing
7.4 Turbulent Diffusion and Radio-Astronomy
7.5 Helicopter Rotor and Satellite Orbit Stability
7.6 Biological Waste Treatment, Hydrology and Air Quality
7.7 Seismology and Structural Mechanics
7.8 Fatigue Cracking, Optical Bistabilityand Nemantic Liquid Crystals
7.9 Blood Clotting Dynamics and Cellular Energetics
7.10 Josephson Tunneling Junctions Communications and Stochastic Annealing
PART IV. Time Discrete Approximations
8. Time Discrete Approximation of Deterministic Differential Equations
8.1 Introduction
8.2 Taylor Approximations and Higher Order Methods
8.3 Consistency, Convergence and Stability
8.4 Roundoff Error
9. Introduction to Stochastic Time Discrete Approximation
9.1 The Euler Approximation
9.2 Example of a Time Discrete Simulation
9.3 Pathwise Approximations
9.4 Approximation of Moments
9.5 General Time Discretizations and Approximations
9.6 Strong Convergence and Consistency
9.7 Weak Convergence and Consistency
9.8 Numerical Stability
PART V. Strong Approximations
10. Strong Taylor Approximations
10.1 Introduction
10.2 The Euler Scheme
10.3 The Milstein Scheme
10.4 The Order 1.5 Strong Taylor Scheme
10.5 The Order 2.0 Strong Taylor Scheme
10.6 General Strong Ito-Taylor Approximations
10.7 General Strong Stratonovich-Taylor Approximations
10.8 A Lemma on Multiple Ito Integrals
11. Explicit Strong Approximations
11.1 Explicit Order 1.0 Strong Schemes
11.2 Explicit Order 1.5 Strong Schemes
11.3 Explicit Order 2.0 Strong Schemes
11.4 Multistep Schemes
11.5 General Strong Schemes
12. Implicit Strong Approximations
12.1 Introduction
12.2 Implicit Strong Taylor Approximations
12.3 Implicit Strong Runge-Kutta Approximations
12.4 Implicit Two-Step Strong Approximations
12.5 A-Stability of Strong One-Step Schemes
12.6 Convergence Proofs
13. Selected Applications of Strong Approximations
13.1 Direct Simulation of Trajectories
13.2 Testing Parametric Estimators
13.3 Discrete Approximations for Markov Chain Filters
13.4 Asymptotically Efficient Schemes
PART VI. Weak Approximations
14. Weak Taylor Approximations
14.1 The Euler Scheme
14.2 The Order 2.0 Weak Taylor Scheme
14.3 The Order 3.0 Weak Taylor Scheme
14.4 The Order 4.0 Weak Taylor Scheme
14.5 General Weak Taylor Approximations
14.6 Leading Error Coefficients
15. Explicit and Implicit Weak Approximations
15.1 Explicit Order 2.0 Weak Schemes
15.2 Explicit Order 3.0 Weak Schemes
15.3 Extrapolation Methods
15.4 Implicit Weak Approximations
15.5 Predictor-Corrector Methods
15.6 Convergence of Weak Schemes
16. Variance Reduction Methods
16.1 Introduction
16.2 The Measure Transformation Method
16.3 Variance Reduced Estimators
16.4 Unbiased Estimators
17. Selected Applications of Weak Approximations
17.1 Evaluation of Functional Integrals
17.2 Approximation of Invariant Measures
17.3 Approximation of Lyapunov Exponents
Solutions of Exercises
Bibliographical Notes
Bibliography
Index
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