Variability of paths and differential equations with $BV$-coefficients

We define compositions $\phi (X)$ of H"older paths $X$ in ${\mathbb{R}}^n$ 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 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.