Constrained non-linear estimation and links with stochastic filtering

Hasnaa Zidani (LMI); Laurent Mertz (CUHK); Louis-Pierre Chaintron (DMA); Philippe Moireau (CMAP)

REVIEW 3 major objections 4 minor 9 references

Constrained non-linear estimation and links with stochastic filtering

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A reflected, non-smooth constrained observer is characterized by an HJB equation with mismatched boundary conditions, and a small-noise stochastic filter concentrates on its minimizers.

desk verdict The HJB boundary-condition analysis is solid and new, but the advertised small-noise filtering limit only holds for frozen smooth observation paths, not for the actual Brownian observation process. read the letter →

arxiv 2502.01200 v1 pith:DGMWW6GC submitted 2025-02-03 math.OC

Louis-Pierre Chaintron (DMA) , Laurent Mertz (CUHK) , Philippe Moireau (CMAP) , Hasnaa Zidani (LMI) This is my paper · ORCID

classification math.OC MSC 49L2593E1160F1049J40

keywords constrainedestimationsweepingprocessviscositysolutionNeumannboundaryconditionstochasticfilteringlargedeviationsLaplaceprincipleMortensenobserver

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies how to estimate the hidden state of a dynamical system that is forced to stay inside a bounded convex domain by a normal-cone reflection, when noisy observations arrive in real time. The central result is that the deterministic cost-to-come function for this non-smooth constrained dynamics is a viscosity solution of a Hamilton-Jacobi-Bellman equation, but its sub-solutions and super-solutions obey different Neumann boundary conditions; uniqueness holds only under an inward-pointing drift condition. The paper then proves a Laplace principle connecting this deterministic value function to stochastic filtering: as the process and observation noise vanish, the reflected filter density concentrates on the minimizers of the cost-to-come, which is precisely the Mortensen observer. Because the reflecting dynamics are well posed only forward in time, the usual time-reversibility argument used in smooth filtering is unavailable; the proof works around this with a duality identity. If the result is right, deterministic constrained observers and small-noise stochastic filters are asymptotically equivalent for a broad class of non-smooth systems.

What carries the argument

The central mechanism is the logarithmic transformation that turns a linear parabolic equation into a nonlinear HJB equation: with q~ε the solution of the no-flux Zakai-type problem (24), define V^ε = -ε log q~ε. Under the classical existence result [Fri08], V^ε solves the HJB equation with boundary condition b·n + (1/2)∂V^ε/∂n = 0. A duality identity ∫ Φ^ε(0,x) q~ε(t,x)dx = ∫ Φ^ε(t,x) q~ε(0,x)dx from [Huc90] links the filter density to the solution of the backward dual parabolic problem, and a verification argument gives a control representation of V^ε with reflected controlled dynamics. Comparing V^ε with $V^{0}$ yields Lemma 4.1 and then the Laplace principle. Viscosity solutions here are understood in the Barles-Lions Neumann sense, where the boundary condition is relaxed by allowing the equation to hold at the boundary.

What would settle it

A concrete check: take G=(0,1), drift b(x)=1, observation h=0, observation rate ˙y≡0, and initial log-density ψ(x)=(x-1/2)^2/2; solve the one-dimensional no-flux parabolic problem (24) exactly by spectral expansion and compare the left side of Theorem 2.8 with inf_{x∈[0,1]}(Φ(x)+V(t,x)) for a nonconstant continuous Φ such as Φ(x)=x. If -ε log ∫ $e^{{-Φ/ε}}$ q~ε(t,x)dx fails to approach inf(Φ+V), or if the limit instead equals the infimum of a function satisfying only the 1/2 super-solution boundary condition, the Laplace principle as stated is false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is twofold. First, the constrained cost-to-come V defined by the infimum over square-integrable disturbances that drive the reflected trajectory to x at time t satisfies the HJB equation (14) in G, with the sub-solution boundary condition b·n + ∂V/∂n = 0 and the super-solution boundary condition b·n + (1/2)∂V/∂n = 0 on ∂G (Theorem 2.5). Since these boundary conditions differ, a comparison principle is not proved in general; however, when b·n ≤ 0 on ∂G the paper obtains uniqueness and recovers the state-constrained viscosity solution of [CL90]. Second, for the reflected diffusion with small noise √ε, the normalized filter density obeys the Laplace principle -ε log ∫ $e^{{-Φ/ε}}$ q~ε(t,x)dx → inf_x(Φ(x)+V(t,x)) for continuous Φ (Theorem 2.8), showing that the stochastic filter concentrates on the same minimizers as the deterministic observer. A quantitative convergence result, |V^κ - V| ≤ $Cκ^{{-1/4}}$, justifies replacing the non-smooth inclusion by a penalized smooth dynamics.

Load-bearing premise

The load-bearing premise is that the reflected filter density solves the classical no-flux parabolic problem (24) with enough boundary regularity, $C^{1}$ in time and $C^{2}$ in space away from t=0, and that the duality identity from [Huc90] applies; if the density develops singularities at the boundary, the log-transform and the verification argument connecting the stochastic filter to V collapse.

Editorial extensions

If this is right

Under an inward-pointing drift, the constrained HJB equation (14)-(16) has a unique viscosity solution, so the deterministic observer is well-defined and can be computed by standard numerical schemes for viscosity solutions.
The reflected filter's conditional density satisfies a large deviation principle with rate function x ↦ Φ(x)+V(t,x), so in the small-noise limit the optimal estimator is the minimizer of the cost-to-come, extending the Kalman-Bucy concentration to non-smooth constrained dynamics.
The O(κ^{-1/4}) uniform convergence of the penalized value function gives a quantitative justification for approximating the reflecting inclusion by the smooth penalized dynamics in numerical implementations.
The boundary-condition mismatch (the factor 1/2 in the normal derivative) is not an artifact: it reflects the fact that reflected diffusions spend zero Lebesgue time on the boundary while constrained deterministic trajectories can slide along it; any numerical filter must respect this asymmetry near ∂G.
If b·n ≤ 0 fails, the deterministic cost-to-come may not be the unique solution of the HJB problem, so the equivalence established here relies on the drift direction at the boundary; practical observers in outward-drift domains should be treated as a separate regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension the authors leave implicit: the same Laplace principle should hold for the normalized conditional density π^ε, because the normalizing constant is independent of x and drops out of the infimum; this would make the concentration statement directly usable for particle-filter or ensemble Kalman initializations.
A sharper rate for the penalization convergence is plausible: replacing the crude distance estimate of Lemma 3.1 by a refined reflection-coupling inequality may improve κ^{-1/4} to κ^{-1/2}, testable in one dimension where the penalized and reflected trajectories can be computed explicitly.
The duality technique, which bypasses the missing comparison principle, could be transferred to sweeping processes with time-dependent constraint sets G(t) or to backward reachability problems, where non-reversibility similarly prevents a direct HJB comparison.
The factor-of-two boundary asymmetry predicts that for outward-pointing drift the small-noise filter may be better approximated by the super-solution boundary condition than by the sub-solution one; this could be checked numerically on a half-space example and, if confirmed, would guide filter design in constrained problems.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: Theorem 2.5 is a genuine new result and the proof looks sound. Theorem 2.8 is not, as stated, a result about the stochastic filter—it is a Laplace principle for a deterministic PDE with a frozen C1 observation path, and the paper does not bridge the gap to Brownian observations.

What is actually new and valuable: the two different Neumann boundary conditions for subsolutions and supersolutions of the constrained cost-to-come, and the comparison/uniqueness under b·n ≤ 0. That is a real advance over the earlier one-dimensional half-line setting. The quantitative penalisation convergence in Theorem 2.2 is also carefully done, and the Section 3 proofs are reasonably detailed.

Soft spots. First, the technical assertion in the subsolution proof that one can find admissible trajectories with prescribed terminal control, including at the boundary, is stated without proof or reference. It may be true, but it needs a lemma. Second, the numerical illustration promised in the introduction is not in v1; that is minor. Third, and more important, Theorem 2.8 is oversold. The derivation fixes a C1 realisation y of the observation process. Brownian observation paths are almost surely nowhere differentiable, so the q̃ε in the theorem is not the conditional density for any actual observation path. The paper gives no argument, through robust filtering, joint large deviations, or path approximation, that the Laplace principle transfers to the real filtering problem. The h=0 case makes the gap visible: the true filter is independent of Y and concentrates according to the signal's own LDP, while the PDE rate function still contains ∫|dot y|². So the theorem is a correct statement about a controlled PDE, but not the stochastic filtering connection advertised in the abstract. The imported regularity from [Fri08] and duality from [Huc90] are probably fine for the PDE reading; they are not the issue.

Overall, the HJB part deserves a serious referee and is likely publishable. The filtering part needs either a real transfer argument or an honest reframing. I would send it to review and ask the authors to address the path-regularity gap; as it stands, I would not repeat the filtering claim in print.

Referee Report

3 major / 4 minor

Summary. The paper considers deterministic minimum-energy state estimation for the reflected differential inclusion (1) in a bounded convex domain G. It defines the cost-to-come V in (10), proves a quantitative convergence rate κ^{-1/4} for a Moreau–Yosida penalised estimator (Theorem 2.2), and characterises V as a viscosity solution of the HJB equation (14) with different Neumann-type boundary conditions for sub- and super-solutions, with uniqueness when b·n≤0 (Theorem 2.5). It also states a Laplace principle (Theorem 2.8) claiming that the small-noise reflected filtering density concentrates on minimisers of V, via a duality argument for the no-flux PDE (24).

Significance. The deterministic results are a substantial extension of earlier work [Cha+23; JB88a; Fle97] to reflected dynamics: the explicit convergence rate, the asymmetric boundary conditions, and the uniqueness under an inward-pointing condition are valuable, and the viscosity proofs are detailed. However, the advertised link to stochastic filtering is not proven: Theorem 2.8 concerns the deterministic PDE (24) with a frozen C1 observation path, not the conditional law of the filter (17)-(18). Since actual observation paths are a.s. nowhere differentiable, and for h=0 the filter is independent of Y while the PDE limit contains ∫|ẏ|^2, the stochastic claim is currently unsupported. The deterministic part remains a solid contribution, but the title and abstract overstate the achieved connection.

major comments (3)

[Section 2.3.3, Theorem 2.8] The theorem is a statement about the deterministic no-flux problem (24) driven by a fixed C1 path y, and the proof in Section 4 uses only the PDE (24) and the duality identity (34). The observation process in (18) does not have C1 paths, so this theorem does not describe the conditional law of the stochastic filter. The sentence after Theorem 2.8 ('As ε→0, this tells that the non-normalised density q̃^ε concentrates...') is not justified, and the claim in the section introduction that π^ε_t concentrates on minimisers of V is not proved. Moreover, for h=0 the filter is independent of Y, whereas the limit of the PDE expression depends on (1/2)∫|ẏ|^2; hence the two limits cannot coincide without an additional regularisation or joint large-deviation argument. The paper must either prove a Laplace principle for the robust/pathwise Zakai equation valid for arbitrary continuous observation paths, or explicitly restrict the announced stochastic filtering claim.
[Section 3.2, proof of Theorem 2.5(i)] The assertion that for every ᾱ∈R^n there exists an admissible pair (x_ω(0),ω) ∈ A^G_{t,x} with continuous ω(t)=ᾱ and x_ω(s)∈G for s<t, and for x∈∂G under [b(t,x)+ᾱ]·n(x)≥0, is used to derive the sub-solution property by taking ᾱ=∇φ(t,x), but no proof is given. This is a nontrivial controllability statement for a sweeping dynamics with unilateral constraint; it should be stated and proved as a lemma. Without it, the sub-solution part of Theorem 2.5 is incomplete.
[Section 4, Lemma 4.1] The proof cites [PR14, Proposition 4.16-I/II] for moment bounds on reflected SDEs, but the passage from these process estimates to the claimed uniform bound sup |V^ε_Φ - V^0_Φ| ≤ C ε^{1/4} is not fully justified. The infimum in (35) is over adapted control processes, while V^0_Φ is a deterministic optimal control value; comparing the two requires an argument (e.g., a measurable selection or a dynamic programming estimate) that (35) is stable under ε→0. The one-sentence 'we then plug this into the minimisation' with a Cauchy–Schwarz bound is insufficient as written.

minor comments (4)

[References] The bibliography lists the identical reference as [JB88a] and [JB88b]; these should be consolidated into one entry.
[Section 2.3.3] In the display of the observation process, 'h(t,X^κ,ε_t)' should read 'h(t,X^ε_t)'.
[Proof of Theorem 2.2] The citation 'Lemma 3.2-(i)' is incorrect: Lemma 3.2 is a single statement, and the boundedness of V^κ used there follows from Lemma 3.1.
[Theorem 2.8 and its proof] The codomain of Φ is written as 'Φ : G → 0'; this should be 'Φ : G → R'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central results are derived from explicit estimates and external PDE/SDE theorems, with self-citation used only as background.

full rationale

The derivation chain is self-contained. The value function V is defined independently as the infimum in (10) over the non-smooth dynamics, and Theorem 2.5 characterises it as a viscosity solution of the HJB equation using the dynamic programming principle, direct convex-analysis estimates, and external comparison results from [BL91, CL90]; no fitted parameter or assumed conclusion enters the proof. The penalised approximation V^κ is defined by the same infimum with the smoothed dynamics (12), and Theorem 2.2 is proved from the quantitative estimates in Lemmas 3.1 and 3.2, not from the limit it establishes. For the stochastic link, Theorem 2.8 is not obtained by postulating the answer: the density q~ε is introduced as the solution of the no-flux PDE (24), and the Laplace principle is proved through the duality identity (34) from [Huc90], the verification representation (35), and the quantitative convergence of the log-transformed value function in Lemma 4.1, with the final Laplace evaluation imported from the standard result cited in [JB88a, Lemma 6.1]. The only self-citation, [Cha+23], is cited as earlier background and is not used to supply any load-bearing uniqueness theorem or ansatz. I note separately that the C1-realisation freezing in Section 2.3.3 means the stated Laplace principle is proved for the deterministic PDE (24) rather than directly for the rough observation paths of the actual stochastic filter; that is a regularity/transfer gap, not a circular reduction of the result to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters and no new physical or mathematical entities are introduced. The paper relies on standard objects: reflected dynamics, cost-to-come, viscosity solutions, and Zakai equations. The main load is carried by regularity theorems and comparison principles imported from prior literature.

assumptions (6)

standard math The reflected differential inclusion (1) is well-posed for L2 disturbances, with existence and uniqueness of absolutely continuous solutions, from [ET05, Theorem 1].
Invoked in Section 1.1 to guarantee that the controlled dynamics (1) has unique absolutely continuous solutions for every square-integrable disturbance.
domain assumption Uniform invertibility condition (9): sigma(x)^T sigma(x) >= gamma0 Id on G.
Assumed in Section 2 to ensure the admissible set A^G_{t,x} is non-empty; restricts the class of dynamics and observation coefficients.
domain assumption For the HJB and filtering theorems, n = r and sigma = Id.
Explicit restriction in Sections 2.2 and 2.3; the central results are proved only in this isotropic setting.
standard math The no-flux parabolic problem (24) has a classical solution in C([0,T] x G) intersected with C^{1,2}((0,T] x G), from [Fri08, Chapter 5, Corollary 2], and the duality identity (34) from [Huc90, Lemma 3.2] holds.
Load-bearing for Theorem 2.8; if this regularity or duality fails, the Laplace principle proof breaks down.
domain assumption Initial condition for the Laplace principle: sup_x | -eps log q0^eps(x) - psi(x) | tends to 0 and q0^eps is C1 near the boundary.
Assumption in Theorem 2.8; it links the initial filter density to the deterministic initial cost psi.
standard math Viscosity solution framework with Neumann boundary conditions [BL91, Lio85], comparison principle [BL91, Theorem 3], and state-constraint comparison [CL90, Theorem III.2].
Used in Section 3.2 to define and prove uniqueness for the HJB characterization of V.

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Cite this review

Pith. "Pith review of Constrained non-linear estimation and links with stochastic filtering." pith.science (2026). https://pith.science/paper/DGMWW6GC

@misc{pith2026250201200,
  author       = {Pith},
  title        = {Pith review of: Constrained non-linear estimation and links with stochastic filtering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGMWW6GC}},
  note         = {Machine review of arXiv:2502.01200}
}

read the original abstract

This article studies the problem of estimating the state variable of non-smooth subdifferential dynamics constrained in a bounded convex domain given some real-time observation. On the one hand, we show that the value function of the estimation problem is a viscosity solution of a Hamilton Jacobi Bellman equation whose sub and super solutions have different Neumann type boundary conditions. This intricacy arises from the non-reversibility in time of the non-smooth dynamics, and hinders the derivation of a comparison principle and the uniqueness of the solution in general. Nonetheless, we identify conditions on the drift (including zero drift) coefficient in the non-smooth dynamics that make such a derivation possible. On the other hand, we show in a general situation that the value function appears in the small noise limit of the corresponding stochastic filtering problem by establishing a large deviation result. We also give quantitative approximation results when replacing the non-smooth dynamics with a smooth penalised one.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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[Huc90] H.-P. Hucke. “Nonlinear ﬁltering of reﬂecting diﬀu sion processes”. In: Stochastic Processes and their Applications 35.2 (1990), pp. 213–224. [IK96] H. Ishii and S. Koike. “A new formulation of state cons traint problems for ﬁrst-order PDEs”. In: SIAM Journal on Control and Optimization 34.2 (1996), pp. 554–571. [JB88a] M. R. James and J. S. Baras...

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Relaxation of an optima l control problem involving a perturbed sweeping process

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Stochastic partial diﬀerential equat ions and ﬁltering of dif- fusion processes

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[2008]

Hamil ton-Jacobi-Bellman approach for optimal control problems of sweeping processe s

[HPV24] C. Hermosilla, M. Palladino, and E. Vilches. “Hamil ton-Jacobi-Bellman approach for optimal control problems of sweeping processe s”. In: Appl Math Optim 90, 33 (2024). [Hij84] O. Hijab. “Asymptotic Bayesian Estimation of a First Order Equation with Small Diﬀusion”. In...

2024

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Local well-posedness of the Mortensen observer

[BL91] G. Barles and P.-L. Lions. “Fully nonlinear Neumann typ e boundary con- ditions for ﬁrst-order Hamilton–Jacobi equations”. In: Nonlinear Analysis: Theory, Methods & Applications 16.2 (1991), pp. 143–153. [BS23] T. Breiten and J. Schröder. “Local well-posedness of th e Mortensen ob- server”. In: arXiv preprint arXiv:2305.09382 (2023). [CM17] T. H. C...

work page Pith review arXiv 1991

[2] [394]

Optimal control of nonconvex s weeping pro- cesses with separable endpoints: Nonsmooth maximum princi ple for local minimizers

[NZ22] C. Nour and V. Zeidan. “Optimal control of nonconvex s weeping pro- cesses with separable endpoints: Nonsmooth maximum princi ple for local minimizers”. In: Journal of Diﬀerential Equations 318 (2022), pp. 113–

work page 2022

[3] [902]

Nonlinear ﬁltering of reﬂecting diﬀu sion processes

[Huc90] H.-P. Hucke. “Nonlinear ﬁltering of reﬂecting diﬀu sion processes”. In: Stochastic Processes and their Applications 35.2 (1990), pp. 213–224. [IK96] H. Ishii and S. Koike. “A new formulation of state cons traint problems for ﬁrst-order PDEs”. In: SIAM Journal on Control and Optimization 34.2 (1996), pp. 554–571. [JB88a] M. R. James and J. S. Baras...

work page 1990

[4] [1967]

Relaxation of an optima l control problem involving a perturbed sweeping process

19 [ET05] J. F. Edmond and L. Thibault. “Relaxation of an optima l control problem involving a perturbed sweeping process”. In: Math. Program., Ser. B 104 (2005), pp. 347–373. [FK06] J. Feng and T. G. Kurtz. Large deviations for stochastic processes . Ap- plied Mathematical Series

work page 2005

[5] [1970]

Moreau-Yosida regulariz ation of state-dependent sweeping processes with nonregular sets

[JV17] A. Jourani and E. Vilches. “Moreau-Yosida regulariz ation of state-dependent sweeping processes with nonregular sets”. In: Journal of Optimization Theory and Applications 173 (2017), pp. 91–116. [KS68] G. Kallianpur and C. Striebel. “Estimation of stocha stic systems: Arbi- trary system process with additive white noise observation errors”. In: The...

work page 2017

[6] [1977]

Pathwise non-linear ﬁltering

[Dav81] M. Davis. “Pathwise non-linear ﬁltering”. In: Stochastic Systems: The Mathematics of Filtering and Identiﬁcation and Applicatio ns: Proceedings of the NATO Advanced Study Institute held at Les Arcs, Savoie , France, June 22–July 5, 1980 . Springer. 1981, pp. 505–528. [Dem09] A. Dembo. Large deviations techniques and applications . Springer,

work page 1980

[7] [1978]

Stochastic partial diﬀerential equat ions and ﬁltering of dif- fusion processes

1978, pp. 239–269. [Par80] E. Pardoux. “Stochastic partial diﬀerential equat ions and ﬁltering of dif- fusion processes”. In: Stochastics 3.1-4 (1980), pp. 127–167. [Par78b] E. Pardoux. “Filtrage de diﬀusions avec condition s frontieres: caracterisa- tion de la densite conditionnelle”. In: Journées de Statistique des Processus Stochastiques. Springer, 197...

work page 1980

[8] [2006]

Deterministic nonlinear ﬁltering

[Fle97] W. H. Fleming. “Deterministic nonlinear ﬁltering” . In: Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 25.3-4 (1997), pp. 435–454. [FP82] W. H. Fleming and É. Pardoux. “Optimal control for par tially observed diﬀusions”. In: SIAM Journal on Control and Optimization 20.2 (1982), pp. 261–285. [FS06] W. H. Fleming and H. M. Soner. Co...

work page 1997