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

Abstract

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 ${\sigma }_{\epsilon }$. Under the Markovian linear equilibrium, the price-impact coefficient is $\lambda ={\sigma }_v/\sqrt{{\sigma }_u^2+{\sigma }_{\epsilon }^2}$ -- constant in time -- and the cumulative expected transfer from the protocol's liquidity pool to traders over $[0,1]$ is $|{\Pi }_M|={\sigma }_v{\sigma }_{\epsilon }^2/\sqrt{{\sigma }_u^2+{\sigma }_{\epsilon }^2}$. 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.

v4: Framing reconciliation (competition-robust coarse-signal subsidy); explicit per-increment covariance; LVR stated as a structural correspondence (not identity). 14 pages. Third paper in a privacy-subsidy cluster (companions: arXiv:2605.15746, arXiv:2605.19742)