The rough path theory (of Lyons) is a theory of integration (like Riemann's for example) and defines the notion of "rough integral", which can be seen as an extension of the Stieltjes-Young integral. This theory, which combines algebra and analysis, provides a good framework for studying a certain type of deterministic differential equations.
We will start with an introduction to the Stieltjes-Young integral, which defines the integral of a function f (integrand) “against” a function g (integrator), for f and g “with enough regularity”. If V is a Banach space, by analogy with the Stieltjes-Young integral and by following Gubinelli's point of view, we'll define a set of “integrator”: the weakly geometric rough paths. This definition is based on the notion of nilpotent (truncated) free Lie algebra over V. And a set of integrands: the controlled rough paths. This will enable us to define the rough integral of f against g, where f and g are in the sets I have just mentioned.
We will then show how to use rough path theory to study stochastic differential equations (SDEs).
