Variability of paths and differential equations with BV-coefficients

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Abstract

We define compositions φ(X) of Hölder paths X in ℝn 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 ℝn 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.
Original languageEnglish
Pages (from-to)2036–2082
Number of pages46
JournalAnnales de l'Institut Henri Poincaré B, Probabilités et Statistiques
Volume59
Issue number4
DOIs
Publication statusPublished - 4 Nov 2023
MoE publication typeA1 Journal article-refereed

Keywords

  • 111 Mathematics

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