On stochastic stabilization in sample-and-hold sense

Pavel Osinenko, Ksenia Makarova, Aleksandr Rubashevskii

incompletemedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s main claim (practical stochastic stabilization under sample-and-hold from a stochastic Lyapunov pair) is plausible, but the proof as written contains gaps/misuses: (i) it repeatedly asserts the state process is a (local) martingale and then applies Doob’s inequality to ||X_t|| without justification, and (ii) it uses E[L_t] ≤ L̄ to conclude E[||X_t||] ≤ R*, which is not generally valid. These steps can be repaired (e.g., by working with L directly via Dynkin/Markov or using a Khasminskii transform). The candidate solution supplies a rigorous alternative proof under standard conditions, including a clean sampling lemma and a Foster–Lyapunov/Khasminskii argument.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript addresses an important and practically motivated question: how to carry stochastic Lyapunov guarantees through a digital sample-and-hold implementation. The main theorem is valuable and appears essentially correct; however, key parts of the proof misapply martingale tools (claiming the state is a martingale; using Doob on ||X\_t||) and conflate bounds on E[L\_t] with those on E[||X\_t||]. These issues are fixable: one can work with L (or H∘L) directly via Dynkin/optional stopping to establish stability in probability and (r,R)-convergence. Clarifying regularity in u on compacts for the S\&H perturbation estimates would also help. With these revisions, the paper would provide a solid bridge between continuous-time stochastic stabilization and its digital implementation.