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 \begin{equation}\label{path}X_t:=(X^1_t,\ldots,X^d_t)\in\textrm{C}^\alpha([0,T];\mathbb{R}^d)\end{equation} of low \(\alpha\)-Hölder regularity enhanced by its iterated integrals \begin{equation}\label{iterated} \mathbb{X}^{i,j}_t=:\int_0^tX^j_s\circ dX^i_s.\end{equation} 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 \begin{equation}\label{eq} dY_t=f(Y_t)\circ dX_t,\end{equation} 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.