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This chapter defines rough paths. If the path has good smoothness properties, then its chords (the Abelian version of the description) provide an adequate description and can be used to predict its effects on a controlled system. A rough path uses a nilpotent group element, computed using Chen iterated integrals as an extended description. The chapter introduces the notion of a control and proves several basic results for paths whose nilpotent descriptors are appropriately controlled, and gives the formal definition of a rough path. Key theorems are proved. In particular, the extension theorem allowing one to compute all iterated integrals of a rough path, and the notion of an almost multiplicative functional, which is important for the development of an integration theory, are both introduced.

This chapter establishes an integration theory for rough paths. The key result is the existence of the integral of a Lipschitz one-form against a rough path. In particular, this gives a change of variable formula, and allows one to see that the notion of a rough path is a geometric concept, stable under smooth change of variable. A key technical point is the use of the notion of Lipschitz, to be found in Stein's book ‘Singular integrals’, which allows for functions to be Lipschitz of any degree. The basic integration result requires the one-form to be Lipschitz of degree > p-1, in order to integrate to against all p-rough paths.

The key result presented in this book and the result which justifies the definition of a rough path is the universal limit theorem. The chapter proves in section 6.3 that a system of differential equations controlled by a p-rough path has a unique meaning for p-rough paths, providing the system of equations has a Lipschitz smoothness > p. In fact, the proof of this result also shows that the response of the Itô functional to the driving signal is continuous in the p-rough path metric. The proof is a difficult iterative process, but because the bounds are uniform, one sees that the iterations converge uniformly, and so the limit is continuous.

The concept of a differential equation controlled by a rough path can be motivated by quite simple examples. One such example is a linear system driven by two-dimensional noise. This example is developed and an explicit answer is given to the question. Exactly what information should I extract from the driving stimulus or noise in order to accurately predict the response? The notions of a controlled system of Chen's iterated integral are introduced. The main notion in this book is the concept of a rough path. Almost all paths that one encounters in everyday life are only described approximately. Newton observed that a smooth path is actually quite well approximated by its chords. If one wants to describe a path γ over a short time interval from s to t, then it is enough to evaluate γ at these two times and consider the approximation that comes from replacing γ by the straight line with the same increment. This approach is not adequate if the control or path γ is oscillatory on the scale witnessed by the times s and t. If the path γ represented a text, then the chord is simply a word count. It turns out that a better description, one which takes into account the order of the events represented by γ, can be achieved by a description of γ that involves its first few Chen iterated integrals.