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

Abstract

We define compositions $φ(X)$ of Hölder paths $X$ in $\mathbb{R}^n$ 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 $\mathbb{R}^n$ 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.