On invariance of observability for BSDEs and its applications to stochastic control systems

TL;DR

This paper proves that the observability of certain backward stochastic differential equations remains invariant across different probability spaces and shows that key control properties are equivalent under these conditions.

Contribution

It establishes the invariance of observability for observed BSDEs with constant coefficients and links various control properties in stochastic systems across different probability spaces.

Findings

Observability invariance for observed BSDEs with constant coefficients.

Equivalence of weak observability, approximate null controllability with cost, and stabilizability.

Control properties are consistent across various filtered probability spaces.

Abstract

In this paper, we establish the invariance of observability for the observed backward stochastic differential equations (BSDEs) with constant coefficients, relative to the filtered probability space. This signifies that the observability of these observed BSDEs with constant coefficients remains unaffected by the selection of the filtered probability space. As an illustrative application, we demonstrate that for stochastic control systems with constant coefficients, weak observability, approximate null controllability with cost, and stabilizability are equivalent across some or any filtered probability spaces.