An introduction to rough paths

The course will provide an introduction to the theory of rough paths. Loosely speaking, a rough path is a pair $(X_t,{\mathbb{X}}_t)$ which consists of a path $$X_t:=(X_t^1,\dots ,X_t^d)\in {\text{C}}^{\alpha }([0,T];{\mathbb{R}}^d)$$ of low $\alpha$-Hölder regularity enhanced by its iterated integrals $${\mathbb{X}}_t^{i,j}=:{\int }_0^t{X}_s^j\circ d{X}_s^i.$$ Since the iterated integrals on the righthand side of the formula above are not classically defined if $\alpha <\frac{1}{2}$, their values are instead postulated by the generally non-unique matrix ${\mathbb{X}}_t$; such as in the case of a Brownian motion enhanced by its Itô or Stratonovich integrals.

The foremost aim of the course will be to prove the well-posedness of rough differential equations $$d{Y}_t=f(Y_t)\circ d{X}_t,$$ and, in particular, the continuity of the solution with respect to the driving noise $(X_t,{\mathbb{X}}_t)$ as measured by the rough path metric. We will furthermore prove a deterministic Itô formula and Doob-Meyer decomposition for rough paths. Additional topics may include the signature of a rough path and applications to stochastic partial differential equations.

Date and time info
Tuesday 10:30 - 12:00

Keywords
rough path, rough differential equation

Prerequisites
calculus

Audience
MSc students, PhD students, Postdocs

Language
English

Remarks and notes
The course, while self-contained, will draw motivation and examples from probability theory and stochastic processes.