Average preserving variation processes in view of optimization

R\'emi Lassalle

REVIEW 3 major objections 3 minor 47 references

Average preserving variation processes in view of optimization

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Pith's one-line read This paper proves that critical laws of semimartingale actions under average-preserving variations satisfy an Euler-Lagrange equation whose left side is a deterministic process plus a cadlag martingale.

desk verdict New average-preserving least action principle with a real example, but the main theorem outsources its proof to an in-press companion; worth refereeing, but only after the dependency is resolved. read the letter →

arxiv 1908.01641 v2 pith:QIZ7X3FE submitted 2019-08-05 math.PR

Rémi Lassalle This is my paper · ORCID

classification math.PR MSC 60H3093E20

keywords leastactionprincipleaverage-preservingvariationssemimartingalelawsEuler-Lagrangeequationsintrinsiccalculusofforward-backwardSDEsmean-fieldstochasticcontrol

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 establishes a least action principle for functionals of laws of continuous semimartingales, using perturbations that preserve the average of the path and fix both endpoints. It proves that a law is critical for such a variational problem exactly when the Euler-Lagrange expression, formed from the Lagrangian's derivatives with respect to drift and position, equals the sum of a deterministic measurable process and a cadlag martingale. This matters because it turns an infinite-dimensional optimization problem over path laws into a tractable dynamical condition. For the classical quadratic action it connects critical laws to forward-backward systems of mean-field SDEs, and it extends the least action viewpoint to laws that need not be absolutely continuous with respect to a fixed Gaussian reference measure.

What carries the argument

The engine is the interplay of three objects. First, the space of average-preserving variation processes $$\mathcal{A}^{\infty,0}_\nu=\{h\in\mathcal{V}^{\infty,0}_\nu:\mathbb{E}_\nu[h]=0_H\},$$ whose closure in $L^2_a(\nu,H_{0,0})$ is exactly the zero-mean paths in $H_{0,0}$. Second, the intrinsic differential $\delta S_\nu$, defined by differentiating $S$ along the maps $\mathbb{I}_W+\epsilon h$, which makes calculus on laws possible without quasi-invariance. Third, the variational martingale characterization (Lemma 4.1): if $\delta S_\nu$ vanishes on the dense zero-mean variation space, the representing derivative $\xi$ has the form $\xi=\int_0^\cdot A^\nu_s ds+\int_0^\cdot N^\nu_s ds$, with $N^\nu$ a cadlag martingale and $A^\nu$ deterministic. Applying this to the action gradient $\dot{\xi}_t=\partial_v L_t(W_t,v^\nu_t,\alpha^\nu_t)-\int_0^t \partial_x L_s(W_s,v^\nu_s,\alpha^\nu_s)ds$ delivers Theorem 4.1.

What would settle it

Take a smooth Lagrangian and a candidate extremal law $\nu$ satisfying the endpoint and mean constraints, compute $\partial_v L_t-\int_0^t \partial_x L_s ds$ along $\nu$-typical paths, and test whether it can be written as a cadlag martingale plus a deterministic process; one example where this decomposition fails while $\delta S_\nu$ vanishes on all zero-mean variations would refute Theorem 4.1. A more basic check is to verify the density of the variation space in $L^2_a(\nu,H_{0,0})$ for a non-Markovian law, since the proof imports that density from the companion paper.

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Extended reading notes

Core claim

The central claim is Theorem 4.1: for a regular Lagrangian $L$ satisfying the stated growth bounds, the action $S(\nu)=\mathbb{E}_\nu[\int_0^1 L_t(W_t,v^\nu_t,\alpha^\nu_t)dt]$ is intrinsically differentiable at $\nu$, and the first variation $\delta S_\nu[h]$ vanishes for every fixed-endpoint, zero-mean variation $h$ if and only if there are a cadlag martingale $(N^\nu_t)$ and a deterministic measurable process $(A^\nu_t)$ with $$\partial_v L_t(W_t,v^\nu_t,\$\alpha$^\nu_t)-\int_0^t \partial_x L_s(W_s,v^\nu_s,\$\alpha$^\nu_s)ds = A^\nu_t+N^\nu_t,\quad \$\lambda$\text{-a.e.},\ \nu\text{-a.s.}$$ The deterministic process is the signature of the average-preserving constraint: because variations with zero mean cannot shift the overall drift by a constant, the dual variable acquires a time-dependent deterministic component alongside the martingale.

Load-bearing premise

The whole proof depends on the companion framework's claims that the allowed variations are dense enough and that the action is differentiable at the candidate law, together with the growth bounds on the Lagrangian; if any of that fails, the Euler-Lagrange equivalence collapses.

Editorial extensions

If this is right

The Euler-Lagrange equation of Theorem 4.1 gives a concrete necessary and sufficient condition for a semimartingale law to be critical under average-preserving variations.
For the classical action $S(\nu)=\mathbb{E}_\nu[\int_0^1(|v^\nu_t|^2/2+V(W_t))dt]$, criticality is equivalent to the existence of a solution of a forward-backward system of mean-field SDEs in which the backward component appears only through its centered version.
For suitable choices of the action, the extremal laws include laws of continuous semimartingales whose drift characteristic is an integrable process with independent increments.
The average-preserving constraint removes the deterministic drift as an admissible variation, which is why the Euler-Lagrange condition contains the extra deterministic process $A^\nu$ and not only the martingale term.
The one-dimensional example shows that the theorem can be used to build explicit critical laws by solving a stochastic differential equation and reading off the deterministic and martingale components.

Reading between the lines

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

The author does not say this, but the deterministic process $A^\nu$ behaves like a Lagrange multiplier for the mean constraint, so the same martingale-plus-deterministic split should appear in any variational problem over path laws constrained only by mean and endpoints.
Applying the theorem to relative-entropy actions would likely reproduce entropic bridges; such a test would show whether the average-preserving least action gives a new constructive route to the classical bridge problem.
A possible testable extension is to check on simple nonlinear filters whether the extremality condition forces the innovations to be a martingale plus a deterministic process; if it does, the intrinsic calculus offers a variational route into the innovations problem of filtering.
In multi-dimensional systems, $A^\nu_t$ depends only on time, which suggests it can be estimated from sample paths by subtracting the martingale part from the Euler-Lagrange left side and looking for a purely time-dependent remainder.

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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

Colleague,

The one thing to know: this paper gives a genuinely new first-order condition for optimizing over semimartingale laws with a fixed mean, and the one-dimensional example checks out. The second thing: the main theorem is not self-contained. It outsources the differentiability of the action and the variational formula to Theorem 5.1 of the in-press companion [31], by the same author group. You cannot verify the central equivalence from this preprint alone.

What's new: the average-preserving variation class A∞,0_ν, the density result in Proposition 3.2, and the Euler-Lagrange condition with a deterministic process A alongside the martingale term. The link to McKean-Vlasov FBSDEs in Section 5 is also new. Lemma 4.1 is clean and does the work of turning the martingale characterization into the equivalence, once differentiability is granted. Example 5.1 is a real check: the drift Y is explicitly a Brownian motion plus an exponential, and the resulting law is Gaussian with variance 7/3. That is concrete evidence the machinery is not empty.

The soft spots, in proportion. The load-bearing one is the outsourced proof. After defining ξ, the proof says 'the S-differentiability of S follows from Theorem 5.1 of [31]' and 'the proof of the latter also yields' the variational formula. That is exactly the step connecting the Euler-Lagrange expression to the derivative of the action. If Theorem 5.1 requires assumptions beyond (4.12)-(4.13), for example regularity of characteristic maps under (I+εh)_⋆, the proof of Theorem 4.1 fails. This is the core of the paper, not a minor gap. The example does not cover that generality, though it supports the final condition. A smaller point: Proposition 5.1 is stated more than proved; the FBSDE equivalence is asserted rather than demonstrated.

The paper is coherent and honest about its debts; the citation pattern is normal for a companion-paper setup. If [31] is solid, the main result is a useful contribution to stochastic calculus of variations. I would send it to peer review—it deserves referee time—but the verdict should be conditional on the companion framework being available and correct.

For a reader: someone working in stochastic control, Schrödinger problems, or semimartingale optimal transport. I wouldn't cite it in my own work until the dependency is resolved.

Recommendation: accept for peer review, with a clear request that [31] be made available or the needed parts reproduced in an appendix.

Referee Report

3 major / 3 minor

Summary. The paper develops a least-action principle for functionals of laws of continuous semimartingales, where variations are constrained to preserve the average of the Cameron-Martin path (average-preserving variation processes). The main result, Theorem 4.1, characterizes critical points of such action functionals: under regularity and growth conditions on the Lagrangian, a law is critical under all average-preserving variations with fixed endpoints if and only if the Euler-Lagrange expression equals the sum of a cadlag martingale and a deterministic measurable process. Section 5 connects this result to forward-backward SDEs of McKean-Vlasov type and gives a one-dimensional example. The proof relies heavily on the companion paper [31], an in-press article, for the variational framework, differentiability of the action, and the martingale characterization.

Significance. If Theorem 4.1 is valid, it provides a genuinely averaged-preserving Euler-Lagrange condition with an explicit deterministic correction term, which is a nontrivial and potentially useful tool in stochastic control and semimartingale optimization. The one-dimensional example is convincing: solving the SDE gives Y_t = B_t + e^t and X_1 ~ N(e-1, 7/3), matching the stated Gaussian law. The paper also gives a clean statement connecting the variational condition to forward-backward systems. However, the central result is conditional on the framework of [31], which is cited rather than proved, and Proposition 5.1 is stated without proof. The contribution is therefore significant but not yet self-contained.

major comments (3)

[Theorem 4.1, proof (Section 4)] The proof outsources the two load-bearing steps to the in-press companion [31]: the assertion that S is L^2_a(ν,H_{0,0})-differentiable under conditions (4.12)-(4.13), and the formula δS_ν[h] = E_ν[<ξ,h>_H] with ξ defined as in the proof. Both are said to follow from Theorem 5.1 of [31], whose statement and hypotheses are not reproduced. Since the reader cannot check that [31]'s theorem applies to the set S under the stated assumptions, the equivalence (i)⇔(ii) of Theorem 4.1 is not independently verifiable from this manuscript. Please state Theorem 5.1 of [31] in full, verify that conditions (4.12)-(4.13) imply its hypotheses, and give a proof or a published reference for the differentiability formula.
[Lemma 4.1 (Section 4)] Lemma 4.1 also relies on a cited martingale characterization: the proof invokes the variational characterization of martingales 'see [23] or Proposition 1.1 of [31]' and the orthogonality of ∫_0^. (N_t - E_ν[N_0])dt to L^2_a(ν,H_{0,0}). This is a standard result, but the precise version needed for the representation ξ = ∫_0^. A_s ds + ∫_0^. N_s ds, with deterministic A, should be stated explicitly and proved in the present notation, since the pathwise integral of a martingale is not an element of H_{0,0} in general.
[Proposition 5.1 (Section 5)] Proposition 5.1 is asserted without proof. This proposition is the concrete bridge between the variational theory and forward-backward systems, which is advertised as a central output of the paper. The equivalence between the existence of a solution to (5.19)-(5.20) and the variational condition (ii) should be proved directly, or the reduction to Theorem 4.1 should be carried out explicitly, including verification of the regularity conditions (4.12)-(4.13) and the endpoint constraints.

minor comments (3)

[Theorem 4.1, proof (Section 4)] The process ξ defined in the proof as ξ_t = ∫_0^t dot ξ_s ds does not necessarily have ξ_1 = 0, so it need not lie in H_{0,0}. To match the definition of L^2_a(ν,H_{0,0})-differentiability, one should replace ξ by its H_{0,0} projection; the difference is a linear deterministic term that can be absorbed into A^ν, but this step should be stated explicitly.
[Introduction, equation (0.2)] Equation (0.2) writes ∫_0^t ∂_q L(W_s, v^ν_s, α^ν_s), while the theorem uses ∂_x L; the notation in (0.2) appears to be a typo and should be aligned with the rest of the paper.
[Section 5, Proposition 5.1] The integrability condition '∫_0^1 |(σ.σ_t)^{i,j}_s(X)|ds < ∞' is difficult to parse; the notation (σ.σ_t) presumably denotes the matrix σ_s σ_s^T evaluated along X, but this should be written unambiguously.

Circularity Check

2 steps flagged · score 4.0 of 10

Theorem 4.1 outsources S-differentiability and the derivative formula to Theorem 5.1 of the in-press, self-cited companion [31]; Lemma 4.1 likewise imports the martingale characterization from Proposition 1.1 of [31], leaving the central equivalence non-self-contained.

self citation load bearing [Section 4, proof of Theorem 4.1]
"Under those conditions, the S−diﬀerentiability of S follows from Theorem 5.1. of [31]. Moreover, the proof of the latter also yields δSν[h] = Eν [< ξ, h > H], for all h ∈ L2 a(ν, H0,0)."

The equivalence (4.14) iff (4.17) depends on the action S being L2_a(ν,H0,0)-diﬀerentiable and on the explicit derivative formula δSν[h] = Eν[<ξ,h>_H]. Both facts are imported verbatim from [31], an in-press article with overlapping authorship, rather than proved in this paper. The Euler-Lagrange condition (4.17) is not itself assumed in [31], so the claim has independent content; however, the proof's central step reduces to the self-citation and collapses if Theorem 5.1 of [31] is not valid under the stated conditions (4.12)-(4.13).
self citation load bearing [Section 4, proof of Lemma 4.1]
"from the variational characterisation of martingales (see [23] or Proposition 1.1 of [31] for a summary of the proof) and Lemma 2.1 of [31] we obtain the existence of a (F ν t )−martingales (N ν t ) such that (4.7) holds, with Aν t = Eν [ ˙ξt] λ − a.e."

Both directions of Lemma 4.1—the decomposition ξ = ∫A + ∫N and the orthogonality of ∫(N − E[N0])dt to L2_a(ν,H0,0)—are justified by [23]/[31]. Proposition 1.1 and Lemma 2.1 come from the same authors' in-press framework [31]; although [23] gives an external origin for the martingale characterization, the paper does not prove the needed statement itself, so the load-bearing content of the lemma is carried by self-citation. This is a genuine self-containedness gap, not a case of a prediction reducing to a fitted input.

full rationale

The paper does not appear to contain an instance of a fitted parameter being renamed as a prediction, nor does the Euler-Lagrange condition reduce to an identity by definition. The main result, Theorem 4.1, has independent mathematical content: it turns the variational condition δSν[h]=0 on average-preserving variations into the explicit semimartingale decomposition (4.17) with a deterministic process Aν and a martingale Nν. That content is not literally the same as any theorem of [31] quoted here. What raises the circularity score is that the proof of Theorem 4.1 is not self-contained: the S-differentiability of the action and the exact form of its intrinsic derivative are taken from Theorem 5.1 of [31], and Lemma 4.1's martingale characterization is taken from Proposition 1.1 and Lemma 2.1 of [31]. Since [31] is an in-press companion with overlapping authorship, the central derivation chain is supported by an unverified, self-referential citation rather than by a proof contained in this manuscript. The one-dimensional Example 5.1 is worked out and gives some independent support, but it covers only a special case and does not establish the general framework. Therefore the appropriate score is 4: some self-citation is load-bearing, but the central claim still has independent content and no construction-level circularity is exhibited.

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

The paper introduces no fitted parameters or invented entities. It relies on the stochastic calculus framework of [31] and on standard martingale theory. The deterministic process A and the average-preserving variation class are mathematical consequences, not independent postulates.

assumptions (5)

domain assumption The set S of laws of continuous semimartingales with characteristics as in (0.3)-(0.4) is closed under the perturbation maps U_h = I_W + h for h in V^∞,0_ν.
Used throughout Section 4; inherited from [31] and not proven in this paper.
domain assumption The variation process space V^∞,0_ν is dense in L^2_a(ν, H_{0,0}).
Lemma 2.1 of [31], invoked in the proof of Proposition 3.2 and Lemma 4.1 to pass from dense subspaces to all L^2_a.
domain assumption Variational characterization of martingales: if the expected inner product vanishes for all h in the dense space, then the representing process decomposes as a martingale plus a deterministic part.
Proposition 1.1 of [31] and Emery's result [23], used in Lemma 4.1 to obtain the martingale N and deterministic A.
domain assumption The action functional S is L^2_a(ν,H_{0,0})-differentiable at ν under the growth conditions (4.12)-(4.13).
Theorem 5.1 of [31], cited in the proof of Theorem 4.1; the differentiability itself is not proven in this paper.
standard math Standard semimartingale theory, stochastic integration, and Gronwall's lemma are applied without proof.
Background tools used throughout the paper, especially in the example.

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Pith. "Pith review of Average preserving variation processes in view of optimization." pith.science (2026). https://pith.science/paper/QIZ7X3FE

@misc{pith2026190801641,
  author       = {Pith},
  title        = {Pith review of: Average preserving variation processes in view of optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIZ7X3FE}},
  note         = {Machine review of arXiv:1908.01641}
}

read the original abstract

In this paper, we investigate specific least action principles for laws of stochastic processes within a framework which stands on filtrations preserving variations. The associated Euler-Lagrange conditions, which we obtain, exhibit a deterministic process in the dynamics aside the canonical martingale term. In particular, taking specific action functionals, extremal processes with respect to those variations encompass specific laws of continuous semi-martingales whose drift characteristic is integrable with independent increments. Then, we relate extremal processes of classical cost functions, in particular of specific entropy functions, to a class of forward-backward systems of Mckean-Vlasov stochastic differential equations.

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