Stochastic Differential Equations

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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.