Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems

TL;DR

This paper offers a simple proof demonstrating that the invariance of closed sets under stochastic control systems is equivalent to their invariance under deterministic systems, under minimal assumptions.

Contribution

It provides an elementary proof of the equivalence between stochastic and deterministic invariance of closed sets, simplifying previous complex proofs.

Findings

Equivalence of invariance under stochastic and deterministic systems

Minimal assumptions required for invariance equivalence

Simplified proof approach

Abstract

We provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that -- under minimal assumptions -- a closed set is invariant with respect to a stochastic control system if and only if it is invariant with respect to the (associated) deterministic control system.