A stochastic version of Dini’s theorem was found by Itô and Nisio. It provides a powerful tool to deduce the uniform convergence of stochastic processes from their pointwise convergence in Karkhunen-Loeve-type series expansions. Unfortunately, this tool fails in stronger than uniform modes of convergence, such as Lipschitz or phi-variation convergence, the latter mode being natural for processes processes with jumps. In this work we establish a stochastic version of Dini’s theorem given in a new framework that covers processes with jumps and strong modes of convergence.
We apply these results to Lévy driven stochastic differential equations (SDEs) to obtain strong modes of pathwise convergence of approximate solutions to such SDEs. In the process, we extend the celebrated S. J. Taylor's result, on the optimal path variation of Brownian motion, to the case of Lévy processes. Our method uses the continuity of Itô’s map of rough path theory, thus is applicable beyond Lévy processes as a driving noise.
This talk is based on a joint work with A. Basse-O'Connor and J. Hoffmann-Jørgensen.