×

Weak convergence of stochastic integrals and differential equations. I. (English) Zbl 0862.60041

Graham, C. (ed.) et al., Probabilistic models for nonlinear partial differential equations. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo, Montecatini Terme, Italy, May 22-30, 1995. Berlin: Springer. Lect. Notes Math. 1627, 1-41 (1996).
The main topic is the weak convergence of stochastic integrals in the finite-dimensional case, but one can also find here the preliminary knowledge necessary to understand the main topic. The authors give a concise and very nice introduction to semimartingales, stochastic integration, stochastic differential equations and Skorokhod topology. The main problem of the weak convergence is to describe conditions under which the weak convergence of \((X^n,H^n)\) to \((X,H)\) implies the weak convergence of integrals, i.e. convergence of \(\int H^n_-dX^n\) to \(\int H_-dX\). There are considered two conditions: UT (uniform tightness) and UCV (uniformly controlled variations) and their mutual relationships. Each of these two conditions implies the desired result (Theorem 7.10) that \[ \Bigl(H^n, X^n, \int H^n_-dX^n\Bigr)@> d>>\Bigl(H, X, \int H_-dX\Bigr). \] Applications of results on convergence to SDE’s and to numerical analysis of SDE’s are given. A virtue of this paper is a big number of interesting examples. The infinite-dimensional case is considered in Part II (see [Zbl 0862.60042]).
For the entire collection see [Zbl 0844.00022].

MSC:

60H05 Stochastic integrals
60B10 Convergence of probability measures
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)

Citations:

Zbl 0862.60042
PDFBibTeX XMLCite