Rough differential equations containing path-dependent bounded variation terms

Abstract

The existence and uniqueness of strong solutions to reflected stochastic differential equations on domains are established by Lions-Sznitman and Saisho under very general conditions on the boundary. We proved the existence of solutions to reflected rough differential equations under additional stronger conditions on the boundary in a previous paper by extending Davie's numerical scheme. In this paper, we prove the existence of the solutions under the same general condition as in Saisho etal. We prove the existence theorem for general pathdependent rough differential equations. We also study support theorem for the reflected processes.

Càdlàg Rough Differential Equations with Reflecting Barriers

“…So far existence results for reflected RDEs are only known for continuous driving signals, see [BMCR14, Aid15, DGHT19, RTT19]. Similar results have been obtained in the context of sweeping processes with pathwise perturbations [CNM17,FS15b] and pathdependent rough differential equations [Aid16,Ana20], both also covering reflected RDEs.…”

One-dimensional reflected rough differential equations

“…Our aim in this study is to complete the above picture in the one-dimensional situation, that is to prove uniqueness of a solution to the problem (1.2). Actually, for the reader's convenience, we will also provide a detailed proof of the existence of a solution in this setting, and simplify at the same time some of the arguments used by Aida in [1,2]. The subsequent analysis accordingly offers a thorough -and totally self-contained -proof of well-posedness of the problem (1.2).…”

“…The more complex rough (or step-2) version of (1.2), which somehow extends the Brownian model, has been first considered by Aida in [1], and further analysed by the same author in [2] for more general multidimensional domains. Nevertheless, in these two references, only existence of a solution to (1.2) can be established and the uniqueness issue is left open.…”

“…Nevertheless, in these two references, only existence of a solution to (1.2) can be established and the uniqueness issue is left open. The lack of regularity of the Skorohod map clearly appears as the main obstacle towards a uniqueness result in the approach followed in [1,2].…”

On the Rough Gronwall Lemma and Its Applications

“…Thus, the idea is to exhibit a path y that somehow follows the dynamics in (1.1), but is also forced to stay nonnegative thanks to the intervention of some regular local time m at 0. This problem was studied by Aida [1,2], however, only existence of a solution was established and the uniqueness issue was left open. The main difficulty lies in the lack of regularity of the corresponding Skorokhod map which does not allow to treat the problem via the Banach fixed point theorem.…”