Abstract:

We will review an idea due to K. Bichteler who uses a space-time grid to construct the Ito-Integral via a Riemann sum sampled at hitting times (and then stopping times) of the space grid. This concept has been greatly generalized, and has applications in optimal control, deep learning, as well as certain SPDE's.

In the second part, we will review a certain circle of ideas originating from the 'Indian School' based at that time in Strasbourg, where the Ito formula has been derived via the Tanaka formula (normally it is vice-verse), and taken to a countable Hilbertian space structure on distribution space (which is similar to Colombeau algebras).

This will be illustrated with countable, but numeruous applications, and some easy-to-formulate but yet surprisingly unresolved questions.

Everything is based on countless inspiring discussions with Friedrich Hubalek and Paul Eisenberg.