David Jekel, Todd Kemp, and I recently developed a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduced a general theory of stochastic integration and quadratic (co)variation for a certain class of noncommutative processes---analogous to continuous semimartingales---that includes both the $q$-Brownian motions and the classical $n \times n$ matrix-valued Brownian motions. My talk will focus on background and motivation for the approach from both classical stochastic calculus and previous approaches to noncommutative stochastic calculus. David's talk, which immediately follows mine, will explain our constructions and results.

Bio: Evangelos "Vaki" Nikitopoulos is an NSF Graduate Research Fellow at the University of California San Diego (UCSD), under the supervision of Bruce K. Driver and Todd A. Kemp. Before starting his PhD, he received bachelors of science in mathematics and chemistry from Brown University and completed a yearlong Fulbright Scholarship in Budapest, Hungary. His current research focus is noncommutative stochastic analysis with an eye toward applications in random matrix theory, specifically the study of matrix-valued stochastic differential equations.