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

TL;DR

This paper develops a framework for analyzing differential equations driven by H"older paths with coefficients of bounded variation, establishing existence and uniqueness results even with discontinuous coefficients, including applications to fractional Brownian motion.

Contribution

It introduces a new approach to differential equations with BV coefficients driven by H"older paths, extending classical results to discontinuous and irregular coefficients.

Findings

Existence of generalized Lebesgue-Stieltjes integrals for compositions of paths and BV functions.

Existence and partial uniqueness results for differential equations with BV coefficients driven by H"older paths.

Application to equations driven by fractional Brownian motion with discontinuous coefficients.

Abstract

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.