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.
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