Method of Successive Approximations for Stochastic Optimal Control: Contractivity and Convergence

Safouane Taoufik, Badr Missaoui

correcthigh confidence

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

Audit review

The paper proves, under Assumptions III.1 and V.1, that (i) the state map is Lipschitz with kernel M_{μ,T} via the one-sided-Lipschitz condition and σ-Lipschitzness (equation (5) and Theorem III.3), (ii) the adjoint is bounded (Theorem IV.3) and admits a stability estimate when D_xσ is x-independent (Theorem V.2), and (iii) the MSA operator is Lipschitz with constant L_{μ,T} and is a contraction for large μ or small T, yielding geometric convergence (Theorems VI.3–VI.5) . The candidate solution reproduces the same Lipschitz constant L_{μ,T} and contraction regimes, using a different but standard duality argument with the linearized forward flow for the adjoint’s stability. Aside from taking the boundedness of Y as an assumed fact (which the paper proves), the candidate’s reasoning aligns with the paper’s results and is correct.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript establishes contractivity and convergence of MSA for a stochastic control class with one-sided-Lipschitz drift and Lipschitz diffusion. The development—state stability, adjoint boundedness and stability, then MSA contractivity—is logically clean and technically correct. Clarifying the exact assumptions needed for adjoint boundedness, measurability of the selector h, and pointers for some Dini-derivative steps would enhance readability and rigor. These are minor presentation issues; the core results and constants are sound.