Abstract: Publisher SummaryThis chapter discusses the Itô stochastic differential equation. Itô formulated the stochastic differential equation that determines Kolmogorov's diffusion process. The Itô stochastic differential equation is used as a model for a dynamical system perturbed by a special stochastic process with simplifying mathematical properties. This process is called “white noise,” in analogy to white light. The solution of the Itô equation is a stochastic process. Recently, a new input–output point of view has developed in which the Itô equation is regarded as a mapping, taking the input white noise process into an output. In this way any continuous process can be defined without the need for stochastic integration. The so-called robustness properties of stochastic calculus have been a matter of intensive investigation as a means of increasing the applicability of stochastic models as well as for interesting mathematical questions. The stochastic operator point of view described in this chapter offers interesting and different mathematics for stochastic systems characterized by linear or nonlinear stochastic differential operator equations and Volterra integral equations. The chapter also discusses the relations between canonical expansions and the Itô equation. It presents a general expression for the stochastic Green's function and the perturbation limit.
Publication Year: 1983
Publication Date: 1983-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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