  We define compositions $\varphi(X)$ of H\"older paths $X$ in $\mathbb{R}^n$
and functions of bounded variation $\varphi$ under a relative condition
involving the path and the gradient measure of $\varphi$. We show the existence
and properties of generalized Lebesgue-Stieltjes integrals of compositions
$\varphi(X)$ with respect to a given H\"older 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 $\mathbb{R}^n$ driven
by H\"older paths and involving coefficients of bounded variation. Examples
include equations with discontinuous coefficients driven by paths of
two-dimensional fractional Brownian motions.
