Stochastic Differential Equations : With Applications to Physics and Engineering

'Et moi, ..~ si lavait su CO.llUlJalt en revc:nir, One acMcc matbcmatica bu JaIdcred the human rac:c. It bu put COIDIDOD _ beet je n'y serais point aBe.' Jules Verne wbac it bdoup, 0Jl !be~ IbcII _t to !be dusty cauialcr Iabc&d 'diMardod__ The series is divergent; thc:reforc we may be -'. I!.ticT. Bc:I1 able to do something with it. O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world when: both feedback and non­ linearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statcmalts as: 'One service topology has rendered mathematical physics ...•; 'One service logic has rendered c0m­ puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series. This series, Mathematics and Its Applications. started in 19n. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However. the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branc:hes. It also happens, quite often in fact, that branches which were thought to be completely.

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.