Abstract: The stochastic Taylor expansion for SDEs has motivated the study of high order iterated integrals of Brownian motion (e.g. Ben Arous 89′, Kloeden and Platen 92′). The literature has mostly considered the upper bound on the iterated integrals. We obtain a lower bound and a limit theorem for the asymptotic of iterated integrals. The interplay between the probabilistic properties of Brownian motion and the non-commutative structure of iterated integrals gives rise to some intriguing phenomenon. For example, the asymptotic of iterated integrals has a log log improvement in regularity compare to each individual term.
This is part of a wider program to pin down the limiting behaviour of high order iterated integrals, and we will discuss the many open problems in the area.