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Stochastic Differential Equations


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November 2001

Beschreibung

Beschreibung

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Inhaltsverzeichnis

Introduction: Origin of Stochastic Differential Equations.
- I. Stochastic Processes - Short ResumÉ.
- 1. Introductory Remarks.
- 2. Probability and Random Variables.
- 2.1. Basic concepts.
- 2.2. Some probability distributions.
- 2.3. Convergence of sequences of random variables.
- 2.4. Entropy and information of random variables.
- 3. Stochastic Processes - Basic Concepts.
- 4. Gaussian Processes.
- 5. Stationary Processes.
- 6. Markov Processes.
- 6.1. Basic definitions.
- 6.2. Diffusion processes.
- 6.3. Methods of solving the Kolmogorov equation.
- 6.4. Vector diffusion processes.
- 7. Processes With Independent Increments; Wiener Process And Poisson Process.
- 7.1. Definition and general properties.
- 7.2. Wiener process.
- 7.3. Poisson process.
- 7.4. Processes related to Poisson process.
- 8. Point Stochastic Processes.
- 9. Martingales.
- 10. Generalized Stochastic Processes; White Noise.
- 11. Processes with Values in Hilbert Space.
- 12. Stochastic Operators.- Examples.
- II. Stochastic Calculus: Principles and Results.
- 13. Introductory Remarks.
- 14. Processes of Second Order; Mean Square Analysis.
- 14.1. Preliminaries.
- 14.2. Mean-square continuity.
- 14.3. Mean-square differentiation.
- 14.4. Mean-square stochastic integrals.
- 14.5. Orthogonal expansions.
- 14.6. Transformations of second-order stochastic processes.
- 14.7. Mean-square ergodicity.
- 15. Analytical Properties of Sample Functions.
- 15.1. Sample function integration.
- 15.2. Sample function continuity.
- 15.3. Sample function differentiation.
- 15.4. Relation to second-order properties.
- 16. ITÔ Stochastic Integral.
- 17. Stochastic Differentials. ITÔ Formula.
- 18. Counting Stochastic Integral.
- 19. Generalizations.- Examples.
- III. Stochastic Differential Equations: Basic Theory.
- 20. Introductory Remarks.
- 21. Regular Stochastic Differential Equations.
- 21.1. Mean-square theory.
- 21.2. Sample function solutions.
- 21.3. Analysis via stochastic operators.
- 21.4. Asymptotic analysis.
- 21.5. Stationary solutions.
- 22. ITÔ Stochastic Differential Equations.
- 22.1. Existence and uniqueness of a solution.
- 22.2. Relation to Stratonovich interpretation.
- 22.3. State transformations and simple solutions.
- 22.4. Asymptotic properties.
- 22.5. Equations with Markov coefficients.
- 22.6. Equations with jump processes.
- 22.7. Equations with functional coefficients.
- 22.8. Strong and weak solutions.
- 23. Stochastic Abstract Differential Equations.
- 23.1. Introduction; deterministic theory.
- 23.2. Itô stochastic equations.
- 23.3. Other problems.
- IV. Stochastic Differential Equations: Analytical Methods.
- 24. Introductory Remarks.
- 25. Systems with Random Initial Conditions.
- 25.1. Probability distribution of solution.
- 25.2. Liouville equation.
- 26. Linear Systems with Random Excitation.
- 26.1. Solution and its properties.
- 26.2. Stationary solutions; Spectral method.
- 26.3. Nonstationary excitations: random impulses.
- 26.4. Linear systems and normality.
- 27. Nonlinear Systems with Random Excitation.
- 27.1. White noise excitation.
- 27.2. Real random excitation.
- 27.3. Use of maximum entropy principle.
- 28. Stochastic Systems.
- 28.1. General remarks.
- 28.2. Systems with parametric uncertainty.
- 28.3. White noise parametric excitation.
- 28.4. Real random parametric excitation.
- 29. Stochastic Partial Differential Equations.
- 29.1. Use of Hilbert space formulation.
- 29.2. Stochastic KdV equation.
- V. Stochastic Differential Equations: Numerical Methods.
- 30. Introductory Remarks.
- 31. Deterministic Equations: Basic Numerical Methods.
- 31.1. Some approximate methods.
- 31.2. Basic numerical schemes.
- 32. Approximate Schemes for Regular Stochastic Equations.
- 32.1. Method of successive approximation.
- 32.2. Approximation and simulation.
- 33. Numerical Integration of ITÔ Stochastic Equations.
- 33.1. Preliminaries.
- 33.2. Stochastic Euler scheme.
- 33.3. Milshtein scheme.
- 33.4. Stochastic Runge-Kutta schemes.
- 33.5. Multi-dimensional systems.
- 33.6. Approximation and simulation.
- VI. Applications: Stochastic Dynamics of Engineering Systems.
- 34. Introduction.
- 34.1. General remarks.
- 34.2. Underlying models for stochastic dynamics.
- 35. Random Vibrations of Road Vehicles.
- 35.1. On road-induced excitation.
- 35.2. Response to random road roughness.
- 36. Response of Structures to Turbulent Field.
- 36.1. On turbulent-induced excitation.
- 36.2. Random vibrations of elastic plate.
- 37. Response of Structures To Earthquake Excitation.
- 37.1. Description of earthquake excitation.
- 37.2. Stochastic seismic response.
- 38. Response of Structures to Sea Waves.
- 38.1. Description of sea wave excitation.
- 38.2. Ship motion in random sea waves.
- 38.3. Response of offshore platforms.
- 39. Stochastic Stability of Structures.
- 39.1. Stability of column.
- 39.2. Stability of suspension bridge.
- 40. Other Problems.- Appendix..- A.1. Cauchy formula.- A.2. Gronwall-Bellman inequality.- References.

Innenansichten

EAN: 9781402003455
ISBN: 1402003455
Untertitel: With Applications to Physics and Engineering. Softcover reprint of the original 1st ed. 1991. Book. Sprache: Englisch.
Verlag: Springer
Erscheinungsdatum: November 2001
Seitenanzahl: 420 Seiten
Format: kartoniert
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