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

Bao-Zhu Guo, Huaiqiang Yu, Meixuan Zhang

TL;DR

The paper tackles the problem of observability invariance for observed BSDEs with constant coefficients across filtered probability spaces. By developing a backward shift property and linking BSDEs to a parabolic PDE representation, it establishes that observability constants depend only on system structure and time horizon, not on the underlying probability space. It then shows that weak observability, approximate null controllability with cost, and stabilizability are equivalent across spaces, with a Riccati-equation criterion underpinning the algebraic characterization. These results enable translating infinite-horizon stabilizability questions into finite-horizon, cost-based controllability statements, and provide a robust framework for stochastic control under model-space uncertainty. The work blends BSDE/FBSDE theory, PDE representations, and Malliavin calculus to derive invariance results with clear implications for control design under varying filtrations.

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.

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

TL;DR

The paper tackles the problem of observability invariance for observed BSDEs with constant coefficients across filtered probability spaces. By developing a backward shift property and linking BSDEs to a parabolic PDE representation, it establishes that observability constants depend only on system structure and time horizon, not on the underlying probability space. It then shows that weak observability, approximate null controllability with cost, and stabilizability are equivalent across spaces, with a Riccati-equation criterion underpinning the algebraic characterization. These results enable translating infinite-horizon stabilizability questions into finite-horizon, cost-based controllability statements, and provide a robust framework for stochastic control under model-space uncertainty. The work blends BSDE/FBSDE theory, PDE representations, and Malliavin calculus to derive invariance results with clear implications for control design under varying filtrations.

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.

Paper Structure

This paper contains 11 sections, 10 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Backward shift property for BSDEs
  4. A characterization on a special FBSDE
  5. LQ problem on infinite horizon
  6. Control problem and main results
  7. Observability
  8. Approximate null controllability with cost and stabilizability
  9. The proof of Theorem \ref{['yu-theorem-7-24-1']}
  10. The proof of Theorem \ref{['yu-theorem-7-27-2']}
  11. Appendix: Proof of Lemma \ref{['yu-proposition-7-11-1']}

Key Result

Lemma 2.1

Let $k\geq 2$. Suppose that $y_1\in L^2_{\mathcal{F}^w_{kT}}(\Omega,\mathbb{P};\mathbb{R}^n)$, $y\in C_{\mathfrak{T}(w,\mathbb{P})}([0,kT];\mathbb{R}^n)$, $f\in L^2_{\mathfrak{T}(w,\mathbb{P})}(0,kT;\mathbb{R}^n)$ and $\{g_i\}_{i=1}^d\subset L^2_{\mathfrak{T}(w,\mathbb{P})}(0,kT;\mathbb{R}^n)$ satis Define $\widehat{w}(\cdot):=w((k-1)T+\cdot)-w((k-1)T)$ in $\mathbb{R}^+$. Then, $\widehat{y}(\cdot)

Theorems & Definitions (30)

  • Lemma 2.1
  • proof : Proof.
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 2.7
  • Definition 3.1
  • ...and 20 more