The Viability Property for Path-dependent SDE under Open Constraints

TL;DR

This paper investigates the conditions under which a path-dependent stochastic differential equation's solutions remain within a bounded open domain, extending previous invariance results to non-Markovian processes.

Contribution

It extends the invariance property for solutions of path-dependent SDEs to non-Markovian settings with Lipschitz data, broadening the applicability of previous results.

Findings

Extended invariance results to non-Markovian SDEs.

Established viability conditions for open domains.

Generalized previous Markovian invariance theorems.

Abstract

In this note, we study the viability of a bounded open domain in $\mathbb{R}% ^{n}$ for a process driven by a path-dependent stochastic differential equation with Lipschitz data. We extend an invariant result of Cannarsa, Da. Prato and Frankowska [\textit{Indiana Univ. Math. J.} \textbf{59} (2010) 53-78] to a non-Markovian setting.