Rough differential equations with path-dependent coefficients

Abstract

We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical Dupire derivatives.Résumé. -Nous montrons l'existence de solutions aux équations différentielles rugueuses dépendant du chemin avec des coefficients non anticipatifs. Les hypothèses de régularité sur les coefficients sont formulées en termes de dérivées de Dupire horizontales et verticales.

Model‐free portfolio theory: A rough path approach

“…(2018). Indeed, every path‐dependent functionally generated portfolio, which is sufficiently smooth in the sense of Dupire (2019) (see also Cont and Fournié (2010)), is a controlled path and thus an admissible strategy, as shown in Ananova (2020). In the present work, we will principally focus on “adapted” strategies F , in the sense that F is a controlled path, as in Definition 3.3, with $$F_t$$ being a measurable function of $$S|_{[0,t]}$$ for each $$t \in [0,\infty )$$.…”