System Control and Rough Paths

Abstract

Systems evolve and interact, and when they do this in a continuous way, differential equations can be used to provide accurate mathematical models for the interaction or reaction of the controlled system to external stimuli or forcing. If the forcing is not smooth on normal scales, one cannot use classical calculus. The theory of rough paths provides a rigorous mathematical extension of Newtonian calculus, allowing one to model the responses of systems subject to more wildly oscillating or ‘rough’ stimuli. This book defines rough paths as the completion of the piecewise smooth paths under a p-variation rough path metric. Building on the earlier work of K.T. Chen on the iterated integrals of paths, and of Young on the integration of paths with p-variation < 2, the book proves that the Itô functional, taking stimuli to responses, is uniformly continuous in the p-rough path metric; from which it is elementary to make sense of these differential equations when the external stimuli are rough paths. One important initial application of these results is to stochastic differential equations. Almost every Brownian path is a p-rough path for every p > 2. In addition, the theory allows one to consider stimuli outside of the classes traditionally treated by the Itô calculus, for example the book explains how fractional Brownian motion can often be regarded as a rough path. The basic estimates are uniform without regard to dimension, and apply to infinite dimensional noise sources.