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

Technical Summary: The Privacy Subsidy in Continuous-Time Kyle

Problem Statement
This paper addresses the welfare implications of privacy-preserving mechanisms in decentralized finance (DeFi), specifically within the context of Automated Market Makers (AMMs) and order-flow aggregation. While the companion paper [12] established a closed-form "privacy subsidy"—a transfer from a protocol's liquidity pool to traders—in a single-period Kyle model, this work extends the analysis to a continuous-time setting. The core problem is quantifying the cumulative expected loss incurred by a committed Bayesian market maker (MM) who observes aggregate order flow perturbed by independent Gaussian privacy noise. Unlike classical competitive market makers who enforce a zero-profit condition, the model assumes a "committed" pricing rule (typical of smart-contract-based AMMs) where the MM updates prices via Bayes' rule on noisy observations but does not dynamically adjust to ensure zero profit.

Methodology
The analysis employs the continuous-time Kyle–Back framework [1], adapted for privacy-aggregated information.

• Model Primitives: The model features a trading horizon $[0, 1]$, an informed trader observing a normal asset value $v$, noise traders submitting flow $du_t = \sigma_u dW^u_t$, and a privacy channel adding independent Brownian noise $d\varepsilon_t = \sigma_\varepsilon dW^\varepsilon_t$. The market maker observes the perturbed flow $d\tilde{y}_t = dy_t + d\varepsilon_t$.
• Pricing Rule: The MM uses a committed linear pricing rule $dp_t = \lambda_t d\tilde{y}_t$, where $\lambda_t$ is the ex-ante fixed Kalman gain derived from the Bayes projection identity. Crucially, the model does not impose the zero-profit equilibrium constraint found in competitive Kyle models; instead, it accepts the resulting non-zero expected profit (or loss) as the object of study.
• Equilibrium Derivation: The authors utilize a linear Markovian ansatz for the insider's strategy ($dx_t = \beta_t(v - p_t)dt$) and solve the associated Hamilton-Jacobi-Bellman (HJB) equation. The solution relies on the standard Kyle–Back rationality condition $\Sigma(1) = 0$ (full information revelation at the horizon) to pin down the equilibrium parameters.
• Welfare Accounting: The cumulative privacy subsidy is derived by decomposing the total expected welfare among the three participants: the informed trader, noise traders, and the market maker. The MM's expected profit is calculated as the residual required to balance the system.

Key Contributions and Results
The paper establishes three primary results:

1. Theorem 1 (Markovian Linear Equilibrium): Under the committed Bayesian-AMM pricing rule with privacy noise intensity $\sigma_\varepsilon$, the unique equilibrium features a constant price-impact coefficient $\lambda = \sigma_v / \sqrt{\sigma_u^2 + \sigma_\varepsilon^2}$ and a linearly declining posterior variance $\Sigma(t) = \sigma_v^2(1-t)$. The presence of privacy noise effectively increases the noise trader volatility from $\sigma_u$ to $\sqrt{\sigma_u^2 + \sigma_\varepsilon^2}$ in the equilibrium conditions.

2. Theorem 2 (Cumulative Privacy Subsidy): The paper derives a closed-form expression for the cumulative expected transfer from the liquidity pool to traders (the privacy subsidy) over the horizon $[0, 1]$:
   $$|\Pi_M| = \frac{\sigma_v \sigma_\varepsilon^2}{\sqrt{\sigma_u^2 + \sigma_\varepsilon^2}}$$
   The authors note that this value is exactly twice the single-period analog derived in [12]. They clarify that this "doubling" is a standard feature of the Kyle–Back transition from single-period to continuous-time models (where informed trader profits also double) and is not a specific artifact of the privacy mechanism.

3. Proposition 1 (Structural Duality with LVR): The paper establishes a structural duality between the Loss-Versus-Rebalancing (LVR) identified by Milionis et al. [10] in constant-product AMMs and the privacy subsidy in Kyle markets.

   • LVR quantifies welfare loss due to a mismatch between the committed AMM curve and the external price observation.
   • Privacy Subsidy quantifies welfare loss due to a mismatch between the committed pricing rule and the order-flow observation.
     Both welfare rates factorize into the squared intensity of the relevant noise driver (price diffusion vs. privacy noise) and a function of the committed object. This duality provides a unified framework for calibrating break-even fees in committed-curve exchanges.

Significance and Claims
The paper claims to complete the program of quantifying break-even fees for committed-AMM exchanges operating under privacy-aggregated information environments. Its significance lies in:

• Bridging Literature: It connects the microstructure literature on insider trading (Kyle/Back) with the DeFi literature on AMM welfare losses (LVR), demonstrating that privacy noise creates a welfare gap structurally identical to price observation gaps.
• Operationalizing Privacy Costs: It provides a precise, closed-form metric for the "cost of privacy" in terms of liquidity pool solvency. The authors argue that for a protocol to be solvent, cumulative fee revenue must exceed this cumulative privacy subsidy.
• Design Implications:
  • Time-Invariance: In the continuous-time limit, the cumulative subsidy depends only on the time-averaged variance of the privacy noise, not its temporal profile. This implies that protocols cannot reduce the subsidy by strategically scheduling privacy injection (e.g., weaker privacy near the horizon) without changing the average budget.
  • Fee Calibration: The derived formula offers a direct method for setting proportional fees to cover the privacy subsidy, analogous to how LVR informs fee structures in CFMMs.
  • Discrete vs. Continuous: The authors caution that while the continuous-time model provides a clean closed form, practical implementations (shielded AMMs, MPC engines) must be interpreted as discrete-block deployments where the privacy budget is finite, as a continuous Brownian privacy channel with fixed intensity is incompatible with finite differential privacy guarantees.

The paper concludes by noting that these results hold for Gaussian noise and that extensions to non-Gaussian noise (e.g., Laplace for differential privacy) or jump processes remain open for future work.