×

Lectures on stochastic flows and applications. Delivered at the Indian Institute of Science, Bangalore, under the T.I.F.R.-I.I.Sc. programme in applications of mathematics. Notes by M. K. Ghosh. (English) Zbl 0625.60073

The first part of these lectures is devoted to the study of the basic properties of stochastic flows. In chapter 1 the author describes the Brownian flows by means of their local characteristics: infinitesimal mean and infinitesimal covariance. It is shown that these two objects determine the law of the Brownian flow.
Chapter 2 deals with stochastic differential equations based on C-valued Brownian motions or more generally C-valued continuous semimartingales. It is shown that the equations have a unique solution if their local characteristics are Lipschitz continuous and that the solutions define a stochastic flow. Conversely, under some conditions flows can be expressed as solutions of suitable s.d.e.
The second part of these notes is devoted to limit theorems for stochastic flows. In chapter 3, using a unified method, the author considers following limit theorems: a) Approximation theorems of s.d.e. and stochastic flows; b) Limit theorems for some driven processes; c) Limit theorems for stochastic ordinary differential equations.
Reviewer: A.D.Borisenko

MSC:

60H99 Stochastic analysis
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory