DMS Colloquium: Jan Rosinski

Speaker: Jan Rosinski, University of Tennessee

Title: Series expansions of time-continuous random walks and solutions to random differential equations

Abstract: In 1923 N. Wiener constructed two random trigonometric series which converge uniformly to a limit satisfying conditions for a Brownian motion. The conditions were  earlier  postulated by A. Einstein in terms of partial differential equations.  K. Ito, the founder of Ito stochastic calculus, unified results on the convergence of these and other similar series expansions in the so-called  Ito-Nisio theorem (1968). O. Kallenberg (1974) gave the first generalization of the Ito-Nisio theorem to the uniform convergence of processes with jumps.

Brownian motion describes continuous in space random walk. In this talk we will concentrate on random walks with jumps, their series expansions, and strongest possible modes of their convergence. To this aim we will establish further generalization of the Ito-Nisio theorem.  We will discuss the Ito map, which is just an ODE with a rough input. Using these tools we obtain strong pathwise convergence in numerical solutions of stochastic differential equations driven by Levy processes.

This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen.

Faculty host: Erkan Nane