Privacy Noise and Trader Welfare in Continuous-Time Kyle Markets

Summary

The document extends a single-period Kyle model to a continuous-time setting where a committed automated market maker observes aggregate order flow blurred by independent Brownian noise. In a linear Markovian equilibrium, it derives a constant price-impact coefficient and a cumulative expected transfer from the liquidity pool to traders, both expressed as functions of fundamental-value uncertainty, ordinary order-flow noise, and privacy-noise intensity.

It relates this privacy subsidy to loss-versus-rebalancing, interpreting the two as welfare costs arising from different observation gaps: noisy order-flow observation here and delayed price observation in the comparison. The result is theoretical and rests on the specified equilibrium and noise assumptions; the excerpt provides no empirical test or evidence about how the result performs in actual markets. It frames the analysis as relevant to break-even fee calculations for automated market makers operating with privacy-aggregated information.

Key ideas

• The model adds independent Brownian privacy noise to the market maker’s observation of aggregate order flow.
• In the stated linear equilibrium, price impact remains constant over time.
• The model derives a cumulative expected transfer from the liquidity pool to traders as privacy noise increases.
• The analysis relates privacy-driven welfare costs to loss-versus-rebalancing through their respective observation gaps.

Tags

# The Privacy Subsidy in Continuous-Time Kyle: Cumulative Welfare under Noise-Perturbed Order-Flow Observation


# The Privacy Subsidy in Continuous-Time Kyle: Cumulative Welfare under Noise-Perturbed Order-Flow Observation









We extend the closed-form privacy-subsidy result of Nakamura~(2026, arXiv:2605.15746) from the single-period Kyle model to continuous-time. A committed Bayesian automated market maker observes the aggregate order flow perturbed by an independent Brownian privacy channel of diffusion intensity $σ_\varepsilon$. Under the Markovian linear equilibrium, the price-impact coefficient is $λ= σ_v / \sqrt{σ_u^2 + σ_\varepsilon^2}$ -- constant in time -- and the cumulative expected transfer from the protocol's liquidity pool to traders over $[0,1]$ is $|Π_M| = σ_v σ_\varepsilon^2 / \sqrt{σ_u^2 + σ_\varepsilon^2}$. We then establish a structural correspondence between this cumulative privacy subsidy and Loss-Versus-Rebalancing (Milionis et al.~2022), identifying privacy-noise welfare as the order-flow observation analog of LVR's price observation gap. The result completes the continuous-time Kyle leg of the program of quantifying break-even fees for committed-AMM exchanges under privacy-aggregated information environments.

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