Integration of time-varying cocyclic one-forms against rough paths

Abstract

We embed the rough integration in a larger geometrical/algebraic framework of integrating oneforms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated paths as integrals of one-forms, and demonstrate that they are stable under basic operations.

Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients

“…This can be easily satisfied for the coefficients of (1.2) when 1 3 < α < 1 2 . We note that the recent works Gubinelli, Tindel and Torrecilla [21], and Lyons and Yang [33] have also studied the rough integration for more general integrands.…”

The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations

“…Suppose U, V and W are Banach spaces and p ≥ 1 a real number. We restate the definition of the cocyclic one-form and the dominated path as in [9].…”

“…For p ≥ 1, we denote by [p] the integer part of p. As in [9], we equip the tensor powers of V with admissible norms and assume…”

“…We say ω : {(s, t) |0 ≤ s ≤ t ≤ T } → R + is a control, if ω is continuous, non-negative, vanishes on the diagonal and satisfies ω (s, u) + ω (u, t) ≤ ω (s, t) for 0 ≤ s ≤ u ≤ t ≤ T . As in [9], for…”

Lipschitz-stability of Controlled Rough Paths and Rough Differential Equations

“…In our modest opinion, having the controlled rough path framework properly set-up in full generality along with the key quantitative estimates may also be beneficial and convenient for the broader community. Apart from Lyons' original approach and Gubinelli's controlled path approach, there are numerous other approaches to study differential equations driven by rough paths, some of which further develops the idea of controlled paths (see for instance Davie [Dav08], Gubinelli [Gub10], Hairer [Hai14], Lyons-Yang [LY14]).…”