Leclère and Rosenbaum give traders a mechanism worth taking seriously: everyone can observe and trade the same predictable signal while its direct effect on the observed price disappears exactly. The entire result depends on G(0) = 1/(1−α) for α < 1. Every term in that equality is latent, leaving bar data with no route to calibration.
The equilibrium becomes a kernel equation
The model has one asset, one Brownian driver B and no data at all. Its observed price is S^α = M + P + I^α. The residual martingale M contains price changes that nobody forecasts and is assumed locally ρ-Hölder for every ρ < 1/2. The common forecastable signal P is the Volterra process P_t = ∫₀^t K(t−s) dB_s. The running example uses K(t) = t^{H−1/2} with H ∈ (0, 1/2), giving a rough signal.
Market impact I^α takes propagator form and depends on aggregate position Π^α, with agent identity removed: I^α_t = G(0)Π^α_t + ∫₀^t G′(t−s)Π^α_s ds. Instantaneous impact enters through G(0); the convolution supplies the transient component.
A single number indexes each agent, the forecast horizon λ, distributed under a probability measure ν. Agent λ holds π^{λ,α}t = E[P{t+λ} − P_t + α(I^α_{t+λ} − I^α_t) | F_t]. The model identifies positions with forecasts by fiat. The authors describe this choice as normalizing the common forecast-to-position scale to one. Risk aversion, inventory constraints, cost objectives and optimization are absent. The coefficient α ∈ [0,1] measures the share of anticipated collective impact included in each agent's forecast, and all agents use the same α. The authors mention α(λ) as a natural extension without pursuing it.
Once aggregate positions are restricted to Gaussian-Volterra form, Π^α_t = ∫₀^t φ_α(t−s) dB_s, the equilibrium reduces to a deterministic linear kernel equation: (Id − α D_ν I)φ_α = D_ν K. This reduction drives the paper. What follows is functional analysis on a decaying-function space, followed by small-τ asymptotics for Gamma-smoothed powers.
Everything rests on an exact equality
For the signal-driven price component X^α = P + I^α, the kernel K_α places coefficient A(α, G(0)) = (1 − (1−α)G(0)) / (1 + αG(0)) on the raw signal kernel K. It vanishes exactly when G(0)(1−α) = 1, the paper's rough-signal cancellation condition. Outside that equality, Proposition 5.3 shows that K_α(τ) ~ A·τ^{H−1/2} on the Gamma-regular branch of Definition 5.1. The critical Hölder exponent of X^α remains H < 1/2, so the price retains the signal's roughness.
At α = 1, no finite G(0) meets the condition. Full anticipation of impact therefore cancels nothing by itself; instantaneous impact must equal exactly 1/(1−α).
The condition has to hold exactly.
No force inside the model moves the market toward it.
The paper's strongest result comes at second order. Under cancellation and Gamma(ξ, β) horizons, ξ determines the local regularity of X^α relative to the threshold 1/2 − H. Below the threshold, the price becomes smoother than the original signal yet remains rough, with exponent H + ξ. Above it, the exponent reaches 1/2 whenever the limiting kernel value ℓ_α is nonzero. If ℓ_α = 0 and 0 < ξ < 1, the exponent is min(1, H+ξ). For ξ ≥ 1 with ℓ_α = 0, paths are locally ρ-Hölder for every ρ < 1. The rate β drops out. What matters is the horizon distribution's mass arbitrarily close to zero. A heavy concentration of agents forecasting at vanishing horizons allows roughness to persist in prices.
Aggregate positions remain rough even when prices become Brownian-compatible. Proposition 5.2 states that every Gamma-regular kernel solution inherits the singularity, φ_α(τ) ~ −τ^{H−1/2}/(1+αG(0)), and keeps critical exponent H. Rough order flow can therefore coexist with a smoother price. Signed flow makes that prediction testable.
The Hawkes connection depends on normalization
Section 3.4 matches the model's G to the dimensionless Hawkes propagator Ξ, for which Ξ(0) = 1/(1−‖h‖₁) and Ξ(∞) = 1. With this identification and α < 1, cancellation becomes ‖h‖₁ = α. The authors present the equality as a connection between two a priori distinct forms of endogeneity.
They also state plainly that positions use forecast units while Hawkes flow uses unit trades. Overall impact scales cannot be compared, leaving only the dimensionless kernels available for matching. Footnote 1 goes further. Their scaling family G_α = G/(1−α) differs from the standard nearly unstable Hawkes scaling h_α = αh. Under the latter, the normalized permanent fraction tends to zero rather than remaining fixed. The equality ‖h‖₁ = α consequently relies on one selected normalization and one selected identification.
Approaching the α↑1 boundary
Take G(0)=1 and choose G_α = G/(1−α), preserving cancellation across the family. Given ‖G′‖{L1} < 1, weighted integrability of G′ and the uniform tail bound sup_α ‖Φ_α‖ < ∞, Proposition 4.1 yields Φ_α → −(Id+H)^{-1}K and K_α → 0. Corollary 4.5 then gives sup{0≤t≤T} E[|S^α_t − M_t|²] → 0. The signal-driven part disappears, leaving the price equal to its martingale component in the limit.
The uniform tail bound carries much of the burden. The paper explicitly says the estimate is not automatic. It verifies the condition in an Ornstein-Uhlenbeck benchmark with exponentially decaying transient impact: K(t) = e^{−γt}, G(t) = 1−Θ+Θe^{−κt}, with 0 ≤ Θ ≤ 1/2. According to the authors, the argument also covers positive Laplace mixtures whose decay rates stay bounded away from zero.
Those examples leave out the singular kernels from Section 5, though they do establish compatibility between the tail condition, a nonzero transient component and a selected equilibrium family. The hypothesis is therefore verified for exponential-type benchmarks while remaining open for the fractional kernels behind the regularity results. The authors disclose this gap. Closing it would change the weight I place on the boundary argument.
Equilibrium selection creates another problem. Remark 3.2 gives an explicit failure of uniqueness using deterministic horizon ν = δ_ℓ and constant impact G ≡ c. The homogeneous equation admits f(τ) = e^{κτ}, where κ = (1/ℓ)log(1 + 1/(αc)). With zero exogenous signal, exponentially growing positions sustain themselves and satisfy the stated equilibrium equation.
The paper restores uniqueness by working in Y⁰_K, the space of bounded functions that vanish at infinity together with finitely many horizon-shifted copies of K. Theorem A.6 gives uniqueness under |d_α|‖Q_{c_α}H‖ < 1. Remark 3.4 supplies a concrete version when G is nonnegative, decreasing and convex, with G(0) > 0. The norm has upper bound 2c_αΘ, where Θ = (G(0)−G(∞))/G(0). Uniqueness follows whenever c_αΘ < 1/2, including whenever at least half of instantaneous impact is permanent.
Serviceable, with equilibrium selection assumed. Section 5 adds the Gamma-regular ansatz from Definition 5.1 to the selected branch instead of deriving it from the construction. Footnote 2 describes this as a mild additional regularity assumption, requiring C¹ control solely for the remainder after separating the explicit Gamma-smoothed terms. The requirement remains an assumption.
Can this be fitted?
Our data consist of OHLCV bars. Recovering the aggregate position process Π^α would require signed order flow or trader-level positions sorted by forecast horizon. Estimating the propagator G would also require observing how prices respond to those flows. Daily or minute bars without trade direction or quotes reveal neither object. Since the paper characterizes an equilibrium rather than constructing a signal or execution policy, it provides nothing discrete to run.
Its efficiency claims concern local Hölder exponents and L² convergence of price toward a martingale. They do not address predictability over any horizon traded by a desk, PnL, or whether the cancelled signal had value after costs. Brownian-compatible local regularity can coexist with forecastability over an hour.
One empirical result would change my view. Estimate ξ and H from actual order flow. If participants' horizon distribution lies above 1/2 − H while signed aggregate flow has exponent H and prices do not, the mechanism would have teeth. This horizon estimate is the paper's sharpest empirical implication, and we did not find a test of it. For now, the model gives a well-built account of how efficient prices can coexist with persistent flow, supported by an equality that nothing inside the model enforces.