Areas of areas determine the iterated-integral signature of a path: An updated overview

05.11.2021, 11:00  –  2.22 und online
Arbeitsgruppenseminar Analysis

Rosa Preiss (University of Potsdam)

We consider the anti-symmetrization of the half-shuffle on words, which we call the "area"  operator since it corresponds to taking the signed area of elements of the iterated-integral signature. We show that the iterated application of the area operator is sufficient to recover the iterated-integral signature of a path. Just as the "information"  that the second level adds to the first one is known to be equivalent to the area between components of the path, this means that all the information added by subsequent levels is equivalent to iterated areas.

This talk gives a short review of the main result of our project, as it is stated in the title, and then focuses on applications to martingales, the class of random processes at the center of the field of stochastic analysis, and to piecewise-linear paths. If time permits, I will try to give an insight on how the latter application actually leads towards a first interpretation of the mysterious Tortkara identity the area operation satisfies, first introduced by the Kazakh mathematician Askar Dzhumadil'daev, which can be stated as (ab)(cb)={a,b,c}b, where the tertiary operation is the Jacobinator (which would vanish in a Lie algebra), {a,b,c}=(ab)c+(bc)a+(ca)b.

There will be a  fortnight's pause before we resume the seminar on Friday November 26th with a talk by Claudio Dappiaggi (University of Pavia, Italy).

For those of you who can and would like to join us, please meet us in the seminar Room 2.22 of the maths institute, where we can follow the talk together on screen.

You are welcome to invite your friends and colleagues to join us! If you wish to attend the talks,  please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.