Abstract:
Non-differentiable continuous functions appear in a variety of contexts. For example, the Black-Scholes equations from finance are differential equations driven by a Brownian motion, which itself is an example of a (random) non-differentiable continuous function. To solve these equations, we must be able to integrate against functions which are not even of bounded variation, so that Riemann-Stieljes integration cannot be used.
In this talk, we will give a brief overview of the theory of rough paths which extends the more familiar Riemann-Stieljes and Young integrals to very irregular functions, and allows to treat such differential equations. We will focus on the theory of controlled rough paths introduced by Gubinelli in 2004.
The talk should be accessible to students with a background in real analysis. No knowledge of probability is required.