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60-02 Probability theory and stochastic processes -- Research exposition (monographs, survey articles)

60H15 Probability theory and stochastic processes -- Stochastic analysis -- Stochastic partial differential equations

60H10 Probability theory and stochastic processes -- Stochastic analysis -- Stochastic ordinary differential equations En-ligne: Aperçu Google 1997 | Zentralblatt | MathSciNet

Location | Call Number | Status | Date Due |
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Salle R | 10245-01 / 60 KUN (Browse Shelf) | Available |

This is the first book to expound, in a detailed way, the relation between stochastic flows and stochastic differential calculus. The new points, compared with the many existing treatises on stochastic calculus, are that it emphasizes the study of random flows of diffeomorphisms rather than semimartingales or diffusion processes. One should note that it was only recently observed that the most natural examples of stochastic flows (namely the isotropic ones) could not be described by stochastic differential equations (SDEs) involving a finite number of real-valued Brownian motions. For example, Brownian flows, i.e., flows with independent increments, are described by stochastic differential equations driven by vector-field-valued Brownian motions.

More standard material on Markov processes, tightness and martingales is presented in the two introductory chapters. Then the stochastic integro-differential calculus is developed in terms of flows. Various types of Itô formulas are discussed, as well as some asymptotic properties. Mixing conditions are given for the convergence of flows defined by SDEs driven by vector field-valued semimartingales towards Brownian flows. An approximation theorem and a support theorem are also given.

The final chapter is a study of stochastic partial differential equations. First-order partial differential equations (PDEs) are studied by the characteristic method. This extends to some second-order PDEs with constant second-order coefficients. Applications are given to nonlinear filtering. A convergence theorem for flows is finally obtained. (MathSciNet)

Bibliogr. 340-344. Index

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