Stochastic Differential Equations, 1972 Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge Series, Vol. 72

Stochastic differential equations whose solutions are diffusion (or other random) processes have been the subject of lively mathematical research since the pioneering work of Gihman, Ito and others in the early fifties. As it gradually became clear that a great number of real phenomena in control theory, physics, biology, economics and other areas could be modelled by differential equations with stochastic perturbation terms, this research became somewhat feverish, with the results that a) the number of theroretical papers alone now numbers several hundred and b) workers interested in the field (especially from an applied viewpoint) have had no opportunity to consult a systematic account. This monograph, written by two of the world's authorities on prob­ ability theory and stochastic processes, fills this hiatus by offering the first extensive account of the calculus of random differential equations de­ fined in terms of the Wiener process. In addition to systematically ab­ stracting most of the salient results obtained thus far in the theory, it includes much new material on asymptotic and stability properties along with a potentially important generalization to equations defined with the aid of the so-called random Poisson measure whose solutions possess jump discontinuities. Although this monograph treats one of the most modern branches of applied mathematics, it can be read with profit by anyone with a knowledge of elementary differential equations armed with a solid course in stochastic processes from the measure-theoretic point of view.

I. One-dimensional Stochastic Differential Equations of First Order.- 1. Stochastic Integrals and Differentials.- § 1. The Wiener Process.- § 2. The Stochastic Integral.- § 3. Properties of Stochastic Integrals as Function of the Upper Limit.- § 4. Stochastic Integrals with Random Limits.- 2. The Solutions of Stochastic Differential Equations.- § 5. Stochastic Differential Equations of First Order.- § 6. Existence and Uniqueness of the Solutions.- § 7. Stochastic Equations which Depend on a Parameter.- § 8. Dependence of the Solutions of Stochastic Differential Equations on the Initial Data.- 3. Solutions of Stochastic Differential Equations and Markov Diffusion Processes.- § 9. Markov Processes. Diffusion Processes.- § 10. Diffusion Processes as Solutions of Stochastic Equations.- § 11. Kolmogorov’s Equation.- § 12. Measures in Function Space Induced by Diffusion Processes.- § 13. Formulas for Transition Density Functions.- § 14. Kolmogorov’s Equation for the Transition Probability Density.- § 15. Time-homogeneous Solutions of Stochastic Differential Equations.- 4. Asymptotic Behavior of the Solutions of Stochastic Equations.- § 16. Bounded and Unbounded Solutions of Stochastic Equations.- § 17. Theorems on the Asymptotic Behavior of Solutions.- § 18. Ergodic Theorems.- § 19. Stability of Solutions.- § 20. Some Other Limit Theorems.- 5. Stochastic Differential Equations on a Finite Spatial Interval.- § 21. Boundary Conditions at the Ends of the Interval.- § 22. Processes with Absorption at the Boundary.- § 23. Instantaneous Reflection at the Boundary.- § 24. Delayed Reflection at the Boundary.- § 25. Processes with Jump Reflection at the Boundary.- II. Systems of Stochastic Differential Equations.- 1. Vector Stochastic Differential Equations.- § 1. Stochastic Line Integrals.- § 2. Stochastic Line Integrals as Function of the Upper Limit.- § 3. Stochastic Differential Equations.- 2. Stochastic Differential Equations without After-effect.- § 4. Preliminary Remarks.- § 5. Some Special Types of Stochastic Integrals.- § 6. The Generalized Itô Formula for Stochastic Differentials.- § 7. Stochastic Differential Equations without After-effect.- § 8. Stochastic Differential Equations Depending on a Parameter. Differentiability w.r.t. the Initial Data.- § 9. Solutions of Stochastic Differential Equations as Markov Processes.- § 10. The Distribution of Functional of the Solutions of Stochastic Differential Equations.- § 11. Some Problems Connected with Homogeneous Stochastic Differential Equations.- 3. Asymptotic Behavior of the Solutions of Stochastic Differential Equations.- § 12. Stability of Solutions.- § 13. Boundedness of the Solutions of Stochastic Differential Equations.- § 14. Limit Theorems for Stochastic Differential Equations.