The principal investigator will continue his work in martingale theory and stochastic integration, with emphasis on the vector-valued and multidimensional time set extensions of the classical theory. Martingales, originally introduced as mathematical models of fair games of chance, have found applications in all branches of classical analysis. Two key methods in the classical theory, the use of quadratic variation and of stopping time techniques, break down in the vector- valued and multi-parameter settings, respectively. Thus the challenge for this project is to determine whether the new tools developed within the past decade can serve as effective substitutes.
