Variability of paths and differential equations with BV-coefficients

Abstract

We define compositions φ(X) of Hölder paths X in ℝn and functions of bounded variation φ under a relative condition involving the path and the gradient measure of φ. We show the existence and properties of generalized Lebesgue-Stieltjes integrals of compositions φ(X) with respect to a given Hölder path Y. These results are then used, together with Doss' transform, to obtain existence and, in a certain sense, uniqueness results for differential equations in ℝn driven by Hölder paths and involving coefficients of bounded variation. Examples include equations with discontinuous coefficients driven by paths of two-dimensional fractional Brownian motions.