Rough paths: theory and applications

Zhongmin Qian University of Oxford

In these talks, I will review the theory of rough paths, and present a few its applications.

Itô’s theory of stochastic calculus has been the major achievement in mathematics in the last century. Itô’s theory allows us to treat such differential equations like

dX_{t} = f(t, X_{t})dt + g(t, X_{t})dW_{t}

where (W_{t})_{t≥0} is a Brownian motion. Solutions to this kind of equations are called Itô’s functionals (or Wiener functionals). As a matter of fact, Itô’s functionals are only measurable in general, which causes substantial difficulties in applying non-linear and topological methods to the study of Itô’s functionals. There have been lasting attempts in searching for a right topology with which Itô’s functionals are continuous. Among of them, as far as I understand, the most remarkable one is the topology defined by the pvariation distance, introduced by T. Lyons. The basic idea can be described briefly as the following. The fundamental feature about Brownian motion is that almost all sample paths of Brownian motion are nowhere differentiable, but are α-Hölder’s continuous for any α < 1/2, which implies that almost all its sample paths have finite p-variation on any finite time-interval for any p > 2. That is

∑_{l} |W_{t_{l}} − W_{t_{l−1}} |^{p} < +∞ almost surely.

While, the increment W_{t_{l}} − W_{t_{l−1}} is simply the 1-st order iterated integral

∫_{t_{l−1}}^{t_{l}} dW_{s} .

These facts motivated the definition of p-variation distance among continues paths. Namely, if (X_{t})_{t≥0} is a continuous path in ℝ^{d} with running time [0, T] with finite total variation, then

X_{s,t}^{k} = ∫_{s<t_{1}<···<t_{k}<t} dX_{t_{1}} ⊗ ⋯ ⊗ dX_{t_{k}}

denotes the k-th order iterated integral of the path (X_{t})_{t≥0} over [s, t]. For a fixed constant p ≥ 1 the p-distance between two paths X and Y with finite variations is defined as

d_{p}(X, Y) = max_{k≤[p]} (sup_{D} ∑_{l} |X_{t_{l−1},t_{l}}^{k} − Y_{t_{l−1},t_{l}}^{k}|^{p/k})^{k/p} .

Let GΩ_{p} denote the closure of all continuous paths with finite p-variations under the distance d_{p}, the space of rough paths. In these talks, we present a few examples of rough paths including fractional Brownian motions, stochastic processes with long-time memory, outline a proof of Lyons’ universal limit theorem, and give a few applications.