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

Abstract

We define compositions $\phi (X)$ of Hölder 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 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.

arXiv comment67 pages, 102 references

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• Fractional maximal functions in metric measure spaces
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Cited by in corpus (4)

• Taming singular stochastic differential equations: A numerical method
• Sobolev regularity of occupation measures and paths, variability and compositions
• On the existence and regularity of local times
• Pointwise well-posedness results for degenerate Itô-SDEs with locally bounded drifts