The randomization method in stochastic optimal control

Marco Fuhrman

REVIEW 3 major objections 5 minor 64 references

The randomization method in stochastic optimal control

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This survey shows that the value of a general stochastic optimal control problem is reproduced by a randomized control problem and is given by the time-zero component of the unique minimal solution to a constrained backward SDE.

desk verdict A useful, honest survey of the randomization method; the central theorem is credible, but the proof has two fixable gaps in the appendix. read the letter →

arxiv 2502.06356 v2 pith:YFNL7IEU submitted 2025-02-10 math.OC math.PR

Marco Fuhrman This is my paper · ORCID

classification math.OCmath.PR MSC 60H1093E20

keywords randomizationofcontrolsbackwardstochasticdifferentialequationsconstrainedBSDEsoptimalcontrolHamilton-Jacobi-BellmanfullynonlinearPDEsminimalsolutionsGirsanovtransformation

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 survey establishes that a large class of stochastic optimal control problems can be solved by first randomizing the control, replacing the chosen action process with a piecewise-constant process driven by an independent Poisson random measure, and then letting the control act through Girsanov changes of probability. Its main theorem states that the randomized problem has exactly the same value as the original one, and that this common value is the time-zero component of the unique minimal solution of a constrained backward stochastic differential equation. If correct, this provides a probabilistic representation for general control problems and for fully nonlinear Hamilton-Jacobi-Bellman equations, extending the nonlinear Feynman-Kac formula beyond the semilinear setting.

What carries the argument

The machinery is the randomized control problem: the original control process is replaced by a piecewise-constant process built from an independent Poisson random measure on the action space, and admissible controls become bounded positive intensity fields that reshape the law of the randomizing process via Girsanov transformation while leaving the Brownian motion untouched. The value of this auxiliary problem is then shown to be the first component of the unique minimal solution of a constrained BSDE in which one martingale integrand is required to be nonpositive, with minimality supplying uniqueness and a penalization procedure supplying existence.

What would settle it

A concrete way to test the central claim is to check the omitted density lemma for the stated class of progressive controls on an arbitrary Borel action space: find an admissible control, for instance one with irregular path dependence on the Brownian motion, that cannot be approximated in dt-times-probability measure by any sequence of finite-valued controls measurable at deterministic times. If such a control exists under the paper's assumptions, the equality of the two values would lack its proof and could fail.

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Extended reading notes

Core claim

The central claim is that, under standard Lipschitz and growth assumptions on the coefficients plus a full-support intensity measure on the action space, the value of the original control problem equals the value of the randomized problem, and this common value is represented by the first component at time zero of the unique minimal solution to the constrained BSDE with jumps. The equality is proved by showing the randomized value is no larger than the original value through a pathwise law-matching argument, and no smaller through a density approximation of arbitrary admissible controls by randomized step processes. The survey therefore presents a complete route from a fully nonlinear stochastic optimization problem to a well-posed backward equation, and identifies the constrained BSDE as the common object representing both the control value and the solution of the associated Hamilton-Jacobi-Bellman equation.

Load-bearing premise

The proof that the original value cannot exceed the randomized value rests on a quoted density lemma asserting that simple finite-valued step controls are dense in the space of all admissible progressive controls under an L1-type metric, and the paper does not report that lemma's proof.

Editorial extensions

If this is right

The value of a general stochastic control problem can be computed as the time-zero component of a unique minimal constrained BSDE solution, with no nondegeneracy assumption on the diffusion coefficient.
Fully nonlinear Hamilton-Jacobi-Bellman equations obtain a probabilistic representation analogous to the Feynman-Kac formula, a case previously treated by second-order BSDEs or G-expectation.
The same scheme applies to switching, impulse, stopping, partially observed, mean-field, infinite-dimensional, and jump-diffusion problems, each with its own constrained BSDE.
A randomized dynamic programming principle holds, providing an alternative route to viscosity solutions of the fully nonlinear HJB equation.

Reading between the lines

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

Beyond the paper: the identity suggests a practical duality certificate for optimal control, where approximate minimal BSDE solutions from one side and admissible controls from the other bracket the true value.
Beyond the paper: because the method adds a Poisson clock independent of the original noise, it may yield Monte Carlo schemes for the value function even when the value function has low regularity, replacing PDE solvers with BSDE simulation.
Beyond the paper: applying the same randomization to deterministic optimal control would produce a genuinely stochastic BSDE whose first component at time zero is the value of a deterministic problem, potentially enabling stochastic numerical methods for deterministic control.

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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

Here's the short version: this is a solid survey that does exactly what it says—no new results, but full proofs of the core equivalence and BSDE representation theorems, plus an appendix proof of a result that was previously only in a preprint. It deserves referee time and, after a minor revision, publication. It will be a standard entry point for people who want to learn the method.

What's genuinely useful: the exposition in Sections 4–6 is careful and self-contained up to the quoted density lemma. The survey gathers a scattered literature into one place, and the Appendix finally gives the details of Proposition A.2 (the approximation of an admissible control by a step process with bounded, bounded-away-from-zero compensator density). That alone is a service. The discussion of limitations in Section 2.4 is honest, and the bibliography is comprehensive.

The soft spots are both in the proof of Theorem 4.8. First, Lemma 5.6—the density of simple step controls in the class of progressive controls under the metric ρ̃—is imported from Krylov [49] without proof. It is genuinely load-bearing for the inequality υ0 ≤ υR0, because Proposition A.2 relies on it to reduce an arbitrary admissible control to a finitely-valued, deterministic-grid step process. This is a dependency, not a fatal gap, but for a paper claiming to be a complete exposition it would be better to include a proof or at least a precise statement of the hypotheses.

Second, the appendix contains a wrong estimate. After (A.7), the cumulative sums ∑_{n=0}^{N−1}(λ_{1m}^{−1}+...+λ_{nm}^{−1}) are bounded as if each cumulative sum were just λ_{nm}^{−1}; the actual sum is O(N/m), not O(1/m). The conclusion (A.6) is still repairable—fix N and choose m large enough—but the displayed proof is invalid as written. A referee should ask for that line to be corrected.

The central argument holds up in the sense that both issues are fixable and the underlying theorems are already published. This paper is for readers who want the method in one place, and for researchers who need the appendix proof. I'd send it to review and accept after minor revision.

Referee Report

3 major / 5 minor

Summary. The paper is a survey of the randomization method for stochastic optimal control. It studies a finite-horizon controlled diffusion with Borel control space A under assumptions (A1), constructs an auxiliary randomized control problem driven by an independent marked Poisson process with intensity measure λ, proves the equality of the two values (Theorem 4.8), and shows that the common value is represented by the first component at time 0 of the unique minimal solution to a constrained BSDE with jumps (Theorems 6.2 and 6.3). The proof of the inequality υ0^R ≤ υ0 uses a point-process construction and a Girsanov argument in Section 5.1, while the reverse inequality in Section 5.2 uses a density lemma imported from Krylov, a stability lemma, and a construction deferred to the Appendix. The final sections survey applications to switching, stopping, impulse control, partial observation, McKean-Vlasov equations, infinite-dimensional systems, and other problems, and list open research directions. The manuscript explicitly states that it contains no new results and positions itself as a synthetic exposition.

Significance. If the results are correct, the paper provides a useful rigorous survey of a method that yields BSDE representations for fully nonlinear Hamilton-Jacobi-Bellman equations and for a broad class of stochastic control problems, including non-Markovian and partially observed cases. A particular strength is the detailed treatment of the basic case, including the marked-point-process constructions in Section 3 and the Appendix, which collects material not previously published in this form. The paper is also honest about its scope: it explicitly identifies where proofs are omitted or deferred. However, because the central equality of values rests on load-bearing arguments that are either imported without proof or contain an invalid estimate, the manuscript cannot currently be used as a fully reliable self-contained reference and needs revision.

major comments (3)

[Section 5.2, Lemma 5.6] Lemma 5.6 is the density result quoted from Krylov [49] that AW_0 is dense in AW with respect to the metric tilde-rho. The paper explicitly says the proof will not be reported. This lemma is load-bearing: it is used in Proposition A.2, which in turn underpins the proof of the inequality υ0 ≤ υ0^R in Section 5.2. If the lemma does not apply to the class of FW-progressive controls under assumption (A1), the proof of Theorem 4.8 has a gap. The author should either provide a proof, or state a precise version of the lemma with all hypotheses and a full reference that verifies its applicability to this setting. For a survey that aims at a complete exposition, an unproved external lemma at this critical juncture is a significant omission.
[Appendix, after Eq. (A.7)] The displayed estimate following (A.7) is false. The cumulative sums ∑_{n=0}^{N−1}(λ_{1m}^{−1}+...+λ_{nm}^{−1}) contain N−j copies of λ_{jm}^{−1}, so the correct upper bound is of order N/m, not the claimed ∑_{n≥1} λ_{nm}^{−1} = 1/m. As written, the proof of the claim tilde-rho(bar-alpha, alpha-hat^m) → 0 in (A.6) is invalid. The conclusion is repairable: N is fixed before choosing m, so taking m large with N/m < δ/3 would suffice, but the displayed argument must be corrected. This is not a cosmetic issue, since Proposition A.2 is essential to the proof of υ0 ≤ υ0^R.
[Section 5.2, Lemma 5.7] Lemma 5.7 is the stability result used to conclude that J^R(ν^k) → J(α) from tilde-rho(I-hat^k, alpha-hat) → 0. Its proof is omitted with the remark that it is entirely analogous to Lemma 5.5. However, the lemma is stated with different filtrations G^k and control processes γ^k that are only G^k-progressive, and it is applied to controls that are adapted to different enlarged filtrations. Because this is a load-bearing step in the final convergence argument, a proof or a precise reference should be supplied rather than left as an analogy.

minor comments (5)

[Section 4.5, Theorem 4.8] The statement refers to the 'partially observed control problem', but the problem formulated in Section 4.2 is fully observed; this should be 'original control problem' or 'classical control problem'.
[Section 4.3, Eq. (4.13)] In the product term of κ^{hat-nu}, the symbol appears as ν_{hat-S_n}(hat-eta_n) without the hat; it should be hat-nu_{hat-S_n}(hat-eta_n).
[Section 3.2, Proposition 3.10] In the final displayed formula, the conditional survival probability is written as an integral without the exponential factor; the right-hand side should be exp(−∫∫ ν λ da ds), consistent with the earlier proof and with (3.9).
[Appendix, proof of Proposition A.2] There is a typo 'Fron now on' that should read 'From now on'. Also, in the proof of Lemma A.4 the expression 'H_t = F_t ∨ F^κ_t' should be 'H_t = hat-G_t ∨ F^κ_t' to match the definition of H.
[Section 5.2, Lemma 5.7] In the display of the convergence of the reward functional, the running cost term should read E^Q[∫_0^T f(...) dt + g(...)]; the 'dt' is missing in the displayed formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the survey's value-equality and BSDE-representation proofs are self-contained, and neither external nor self-citations reduce the central claims to their inputs.

full rationale

The paper's load-bearing results are Theorem 4.8 (equality of the original and randomized control values) and Theorems 6.2/6.3 (existence, uniqueness, and representation via the constrained BSDE). Both are proved in the text rather than assumed. The inequality υR0 ≤ υ0 is proved via Lemma 5.3 and Proposition 3.12, with full constructions given. The reverse inequality υ0 ≤ υR0 rests on Lemma 5.6, quoted from Krylov [49] with the proof omitted, and on Proposition A.2, whose proof is then supplied in the appendix. A citation to an external, parameter-free density lemma is not circular: it does not define the target equality into the input, and the paper explicitly reports that the proof is omitted rather than disguising the lemma as its own result. The appendix bound after (A.7) appears to be incorrectly estimated as written, and Lemma 5.6 is a genuinely load-bearing unproved external input; these are correctness and completeness concerns, not circularity. The many self-citations (e.g., [1], [5], [36], [47], [48]) point to prior works in which the same or related results were established, but the present paper reproduces the needed arguments in Sections 3, 5, and 6, so no load-bearing step reduces to an unverified self-citation. There are no fitted parameters, no prediction constructed from its own inputs, and no uniqueness theorem imported from the authors to force the choice of method. The paper explicitly states it contains no new results, consistent with a survey, but the derivation chain it presents is self-contained apart from standard external tools such as Krylov's density lemma and the general theory of marked point processes. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. Its formal claims rest on the explicit assumptions (A1)-(A2) and on standard or quoted background results from stochastic analysis. The randomization process I is a standard Poisson construction, not a new ontological entity.

assumptions (5)

domain assumption Coefficient conditions (A1): Lipschitz continuity in x, linear growth, polynomial growth of f and g, continuity in the control action a.
These are the standing assumptions for the theorems in Sections 4 to 6. They ensure existence and uniqueness of strong solutions and integrability of the reward functional.
domain assumption Existence of a finite measure λ with full topological support on A and a point a0 in A (assumption A2).
Required to construct the randomized process I and to apply the approximation argument in Proposition A.2. The paper notes that such λ always exists and that the value is independent of its choice.
standard math Marked point process calculus: compensator uniqueness, Watanabe's theorem, Girsanov theorem for point processes, and martingale representation for the filtration generated by a Wiener process and a Poisson random measure.
Invoked throughout Sections 3, 5 and 6, for example in Proposition 3.2, Theorems 3.4 and 3.5, and Lemma 6.7.
domain assumption Density lemma (Krylov [49], Lemma 3.2.6): the set of simple controls AW_0 is dense in AW with respect to the metric tilde-rho.
Quoted without proof in Lemma 5.6 and used to prove the inequality υ0 ≤ υR0 in Section 5.2. The reader must consult the reference to verify its hypotheses.
standard math Standard tools of stochastic analysis: Ito formula, Burkholder-Davis-Gundy inequalities, Gronwall lemma, dominated convergence, and weak compactness arguments.
Used in the stability lemmas (5.5 and 5.7), in the uniform estimates for the penalized BSDEs (Lemma 6.10), and in the penalization convergence proof.

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Cite this review

Pith. "Pith review of The randomization method in stochastic optimal control." pith.science (2026). https://pith.science/paper/YFNL7IEU

@misc{pith2026250206356,
  author       = {Pith},
  title        = {Pith review of: The randomization method in stochastic optimal control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFNL7IEU}},
  note         = {Machine review of arXiv:2502.06356}
}

read the original abstract

In this paper we make a survey on the so called randomization method, a recent methodology to study stochastic optimization problems. It allows to represent the value function of an optimal control problem by a suitable backward stochastic differential equation (BSDE), by means of an auxiliary optimization problem having the same value as the starting one. This method works for a large class of control problems and provides a BSDE representation to many related PDEs of Hamilton-Jacobi-Bellman type, even in the fully non linear case. After a general informal introduction we explain the method giving full details in a basic case. Then we try to give a complete picture of the existing applications and we present some related open problems.

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