Stochastic integrals and differential systems

  • M. M. Rao
Part of the Mathematics and Its Applications book series (MAIA, volume 342)


We abstract and extend the stochastic integration with martingale integrators to more general processes for which the dominated convergence theorem is still valid. The motivation here is to obtain a unified treatment of several different stochastic integrals, available in the literature, by means of a generalized boundedness principle based on a fundamental idea formulated by S. Bochner. After presenting the semi-martingale integrals in the next section, to serve as a key example, the desired boundedness principle is treated in detail in Section 2. It is also shown there, and in Section 3, that the earlier integrals fit in this frame work; and several applications are worked out to exhibit the universality of the principle, including some vector and multiparameter cases. The rest of the chapter is devoted to the existence (and unicity) of solutions of both linear and nonlinear higher order stochastic differential equations and its progression to stochastic flows for the L 2,2-bounded case. This work takes up Sections 4 and 5 below, and most of Section 4 appears in book form for the first time. Several other results are included in the Complements section.


Brownian Motion Markov Process Stochastic Differential Equation Differential System Vector Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.University of CaliforniaRiversideUSA

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