Course outline
Overview
The author recently gave a presentation at the European Mathematical Society Congress in Stockholm. The abstract for that talk provides a useful overview setting out the main features of the theory of rough paths. We reproduce it in the next paragraphs.
It is an understatement to say that classical calculus has proved itself a very successful tool for expressing the relationships between evolving systems. Although many of the applications come from Applied mathematics and particularly from control theory, the development of a path into a manifold using a gauge, the development of a path from a manifold into a frame bundle using a connection, and the development of a path in a Lie algebra into a Lie group all provide examples from classical pure mathematics.
However, the classical approach does not easily deal with paths that occur when a random source modifies or perturbs the evolution of a dynamical system. Most of the natural models for noisy evolving systems are far from differentiable. This has stimulated the evolution of stochastic calculus and stochastic analysis with Itô’s theory of stochastic integration, and Doob’s martingale theory, both developed in the middle of the last century, being the decisive tools. Together they have transformed Finance and represent one the most successful areas of Applied mathematics.
However, Itô theory is essentially probabilistic and can only be applied to the sample paths of a special kind of stochastic process called a semi martingale (a condition which, in the one-dimensional case, is broadly equivalent to saying that the process can be re-parameterised to be a Brownian motion).
Motivated by the inroads these and other techniques in stochastic analysis have made in extending our concepts of calculus to non-standard settings, it seems worth reconsidering the possibility that the classical analytic theory could be extended (without probability) so as to provide a theory of differential equations driven by rough paths; a theory rich enough to capture the classical and the probabilistic examples.
In the end, a rather simple idea works. These differential equations obviously have meaning when the input or controlling path is smooth. If the Itô functional takes a smooth control to the resultant evolving state of the nonlinear system, then one can reasonably ask: Are there metrics on smooth paths for which the Itô functional can be shown to be uniformly continuous or even Fréchet differentiable. If there are and if the completions of the smooth paths in these metrics are not complicated then it would seem reasonable to consider these ”generalised paths” as the space in which the paths that drive differential equation should live.
It is a well-known idea, coming from K. T. Chen (geometry) Fleiss (control theory) Platen (stochastic differential equations) and many others, to consider the sequence of iterated integrals of a path in order to obtain a pathwise Taylor series of arbitrary order for the solution to such equations. In effect, they lift the path up into the free nilpotent group of step
K over the original vector space carrying a path (or in Chen’s case, the path is lifted into the space of formal tensor series).It has been shown that Itô functional is uniformly continuous if we consider the holder metric of order 1=p with respect to a non-degenerate homogeneous metric on any of the free nilpotent groups with step K > p - 1. For fixed p, these metrics are all equivalent. These are the rough or generalised paths.
The proof of the continuity theorem has led to a number of different applications. For example, within classical probabilistic theory, it has led to new proofs of the support theorem as well as techniques for proving the existence of solutions to infinite dimensional stochastic differential equations by careful studies of tensor norms on products of Gaussian random variables. It has extended the classical theory, and particularly work of Qian and Coutin allow us to consider stochastic differential equations driven by fractional Brownian motion.
The theorem has also provoked a number of new theoretical questions concerning iterated integrals – even of smooth paths.
Introduction
The good news about rough paths is that this approach provides an effective analytic tool for describing and mathematically modelling the interaction between systems whose evolution is rough. The bad news is that the techniques required to analyse the problem and solve it come from areas of mathematics which, although not strange within their own context, are new for mathematicians working in probability theory. Moreover, the concept of solution that we will introduce, and the idea of a rough path itself both require a significant shift in the way one thinks about "what is a control". One needs to get familiar with the concepts if one is to find the material straightforward. This is particularly the case because most of the core ideas can be thought of in a number of different ways and it helps substantially to be able to move between these different perspectives.
Once the concepts are firmly understood one discovers that many of the significant theorems in stochastic analysis, for example support theorems, are reduced to a very precise and specific questions about classical processes such as Brownian motion. Because these questions are so well focused, it is easy to invest the effort to answer them fully. This can, and in the case of the support theorem does, lead to improved statements of these theorems.
The methods also lead to completely new results: stochastic differential equations can be driven by noise sources that are quite different to the classical semi martingales; there are also questions of a purely analytic nature that surely deserve to be answered.
The introduction is aimed at familiarising the audience with the basic concepts including the definition of a rough path and some historical results motivating and explaining some of the issues that will be central the remainder of the course.
Lecture One
Bounded variation paths. Paths of finite p variation and α - Hölder paths. Lower semi-continuity of these norms. Compactness criteria. Approximation. Brownian motion has finitely variation for all p >2.
Lipschitz functions following the definition in Stein's book on singular integrals including the characterisation in terms of the behaviour of harmonic functions on the half plane. The division and extension properties for Lipschitz functions.
Lipschitz functions of paths of finite p variation. Hölder continuity.
The integral introduced by Young.
Lecture Two
Differential equations driven by rough paths with p variation <2. The existence and uniqueness (and differentiability (rough sketch)) of solutions - following Picard's method.
Mention of counter examples. The non-existence of a bilinear form on any Banach space extending integration from piecewise linear paths to E^2 and carrying Wiener meaure.
Reminder that these techniques are already powerful.
Lecture Three
Linear differential equations driven by paths of finite length - series solutions and motivation for considering iterated integrals.
Factorial decay of the iterated integrals p<2. Do these methods prove that the solution has finite p-variation? Linear differential equations as entire analytic functions.
Applications: rolling a sphere, and solving time inhomogeneous second order odes. Simple data compression.
The abstract object: the full signature of a path segment - as an element of the tensor series. The Chen-multiplicative property.
Linear functionals on the Tensor algebra as co-ordinate iterated integrals on path space. The shuffle product. Group like elements in the tensor algebra.
The range of the signature map is a group. The truncated signature as a map into the free n-step nilpotent Lie group.
The universal properties of the Tensor Algebra.
Lecture Four - The first steps to p>2.
The neoclassical inequality (without proof). A homogeneous metric on the truncated tensor algebra. Paths of finite p-variation (etc.) in the truncated tensor algebra. Brownian Motion with Lévy area as a paths of finite p-variation in the second truncation of the Tensor Algebra.
The unique extension theorem for paths and almost multiplicative functionals. Continuity of the extension process. "Solutions" to linear differential equations. p>2. More entire functions!
Lecture Five - A theory of integration
Weakly geometric, strongly geometric and non-geometric rough paths. Less obvious (but not less interesting) examples of rough paths. Existence and non-uniqueness of geometric rough paths with given projection. The density of the smooth paths?
The line integral of of Lipchitz one forms against a geometric rough path through the construction of an almost multiplicative functional and the use of re-arrangments. A change of variable formulae. Linear images of rough paths.
The almost additive case - other approaches.
Lecture Six - Hard work with a gentle overview to compensate!
A proof of the neo classical inequality. Brutal analysis! Uses the maximum principle for solutions to certain parabolic differential equatons.
Intuition and motivation - abelian versus non-abelian statistics of a control. Universal control - for all (linear) systems.
Lecture 7: Differential Equations make sense!
Giving differential equations meaning! Other equations make sense as well!
Equations are all very well - finding solutions - Use Picard carefully it is not so simple. A dangerous bend in the approximation game.
Rough paths with multiple homogeneity.
Lecture 8: Solutions exist and are continuous functions of the control!
The final steps. Phew!!
Lecture 9: Support theorems other applications
The main theorem has the effect of reducing a number of classical problems to quite explicit hard problems about Brownain motion or some other basic stochastic process.
Because we understand this process so much better we can improve these classical results. The Stroock Varadhan Support theorem is an example.
It also results in our being able to solve differential equations driven by many different stochastic processes - such as fractional Brownian motion and various reversible Markov processes.
David Williams and Thomas Simon have developed the theory for Levy Processes and proved a support theorem for them.
Lecture 10: Data Compression - recovering a path from the signature -
This seems a hard but fascinating problem!
At least for paths of finite length one fully understands the extent to which the signature determines the path. The proof is conceptual with different flavours to the different steps. The theorem clearly a conjecture open for all p>1 which seems a good place to stop.