An Algebraic Approach to Integration of Geometric Rough Paths

Abstract

We build a connection between rough path theory and noncommutative algebra, and interpret the integration of geometric rough paths as an example of a non-abelian Young integration. We identify a class of slowly-varying one-forms, and prove that the class is stable under basic operations. In particular rough path theory is extended to allow a natural class of time varying integrands.Consider two topological groups G 1 and G 2 , and a differentiable function f : G 1 → G 2 . For a time interval [S, T ] and a differentiable path X : [S, T ] → G 1 , the integration of the exact one-form df along X can be defined as:When f and X are only continuous, T r=S df dX r can be defined as f (X S ) −1 f (X T ). Consider a time-varying exact one-form (df t ) t with f t : G 1 → G 2 indexed by t ∈ [0, 1], and X : [0, 1] → G 1 . If the following limit exists in G 2 :