Variability of paths and differential equations with BV-coefficients

We define compositions phi(X) of Holder paths X in Rn and functions of bounded variation phi under a relative condition involving the path and the gradient measure of phi. We show the existence and properties of generalized Lebesgue-Stieltjes integrals of compositions phi (X) with respect to a given Holder 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 Holder paths and involving coefficients of bounded variation. Examples include equations with discontinuous coefficients driven by paths of two-dimensional fractional Brownian motions.