Differential equations driven by rough signals (I): an extension of an inequality of L. C. Young

Abstract

L.C. Young proved that if x t , y t are continuous paths of finite p, p variations in R d where 1 p + 1 p > 1 then the integral t 0 y u dx u can be defined. It follows that if p = p < 2, and f is vector valued and α-Lipschitz function with α > p − 1, one may consider the non-linear integral equation and the associated differential equation: y t = a + t 0 d i=1 f i (y u) dx i u dy t = d i=1 f i (y t) dx i t y 0 = a. (1) If one fixes x one may ask about the existence and uniqueness of y with finite p-variation where to avoid triviality we assume d > 1. We prove that if each f i is (1+α)-Lipschitz in the sense of [7] then a unique solution exists and that it can be recovered as a limit of Picard iterations; in consequence it varies continuously with x. If each f i is α-Lipschitz, one still has existence of solutions, but examples of A.M. Davie show that they are not, in general, unique.