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High-dimensional nonlinear diffusion stochastic processes : modelling for engineering applications /
Annotation This book is one of the first few devoted to high-dimensional diffusion stochastic processes with nonlinear coefficients. These processes are closely associated with large systems of Ito's stochastic differential equations and with discretized-in-the-parameter versions of Ito's...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore ; River Edge, NJ :
World Scientific,
2001.
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| Colección: | Series on advances in mathematics for applied sciences ;
v. 56. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Contents; Chapter 1 Introductory Chapter; 1.1 Prerequisites for Reading; 1.2 Random Variable. Stochastic Process. Random Field. High-Dimensional Process. One-Point Process; 1.3 Two-Point Process. Expectation. Markov Process. Example of Non-Markov Process Associated with Multidimensional Markov Process; 1.4 Preceding Subsequent and Transition Probability Densities. The Chapman-Kolmogorov Equation. Initial Condition for Markov Process; 1.4.1 The Chapman-Kolmogorov equation; 1.4.2 Initial condition for Markov process.
- 1.5 Homogeneous Markov Process. Example of Markov Process: The Wiener Process1.6 Expectation Variance and Standard Deviations of Markov Process; 1.7 Invariant and Stationary Markov Processes. Covariance. Spectral Densities; 1.8 Diffusion Process; 1.9 Example of Diffusion Processes: Solutions of Ito's Stochastic Ordinary Differential Equation; 1.10 The Kolmogorov Backward Equation; 1.11 Figures of Merit. Diffusion Modelling of High-Dimensional Systems; 1.12 Common Analytical Techniques to Determine Probability Densities of Diffusion Processes. The Kolmogorov Forward Equation.
- 1.12.1 Probability density1.12.2 Invariant probability density; 1.12.3 Stationary probability density; 1.13 The Purpose and Content of This Book; Chapter 2 Diffusion Processes; 2.1 Introduction; 2.2 Time-Derivatives of Expectation and Variance; 2.3 Ordinary Differential Equation Systems for Expectation; 2.3.1 The first-order system; 2.3.2 The second-order system; 2.3.3 Systems of the higher orders; 2.4 Models for Noise-Induced Phenomena in Expectation; 2.4.1 The case of stochastic resonance; 2.4.2 Practically efficient implementation of the second-order system.
- 2.5 Ordinary Differential Equation System for Variance2.5.1 Damping matrix; 2.5.2 The uncorrelated-matrixes approximation; 2.5.3 Nonlinearity of the drift function; 2.5.4 Fundamental limitation of the state-space-independent approximations for the diffusion and damping matrixes; 2.6 The Steady-State Approximation for The Probability Density; Chapter 3 Invariant Diffusion Processes; 3.1 Introduction; 3.2 Preliminary Remarks; 3.3 Expectation. The Finite-Equation Method; 3.4 Explicit Expression for Variance; 3.5 The Simplified Detailed-Balance Approximation for Invariant Probability Density.
- 3.5.1 Partial differential equation for logarithm of the density3.5.2 Truncated equation for the logarithm and the detailed-balance equation; 3.5.3 Case of the detailed balance; 3.5.4 The detailed-balance approximation; 3.5.5 The simplified detailed-balance approximation. Theorem on the approximating density; 3.6 Analytical-Numerical Approach to Non-Invariant and Invariant Diffusion Processes; 3.6.1 Choice of the bounded domain of the integration; 3.6.2 Evaluation of the multifold integrals. The Monte Carlo technique; 3.6.3 Summary of the approach; 3.7 Discussion.