Todd Kemp, Vaki Nikitopoulos, 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. Vaki's talk, which immediately precedes mine, focuses on background and motivation for the approach. My talk will explain our constructions and results, including our noncommutative It\^{o}'s formula.

Bio: David Jekel is a mathematician from the United States studying von Neumann algebras, and especially free probability, random matrix theory, and model theory of tracial von Neumann algebras. He finished his Ph.D. in 2020 at UCLA with Dima Shlyakhtenko, then did an NSF postdoc at UCSD with Todd Kemp. Currently, he is a postdoc at the Fields Institute working with Ilijas Farah.