Private Estimation of Networked Ornstein–Uhlenbeck Processes with Discrete Observations

We study privacy-preserving parameter estimation for networked Ornstein–Uhlenbeck (OU) processes with known topology and a fixed sampling interval under event-level differential privacy. Unbounded stationary Gaussian states yield infinite global sensitivity for the empirical moment queries. We therefore clip each sampled state to a Euclidean ball and jointly privatize the lag-zero and lag-one empirical covariance moments with a Gaussian mechanism. Positive-definite projection and OU-specific spectral stabilization then restore the domain required by the inverse-square-root and matrix logarithm reconstruction. Positive definiteness of the unrestricted reconstructed drift may be lost after the structured coefficients are extracted. We therefore add a final constrained structural projection that ensures that the matrix θ^nIN−β^nA is stable in finite samples. For fixed network dimension and sampling interval, we establish consistency and joint asymptotic normality. The limiting covariance decomposes into the long-run sampling variation of the serially dependent process and a privacy-induced term κ2JJ⊤, which vanishes when κ=0. Monte Carlo experiments are consistent with this decomposition, quantify the finite-sample privacy–utility tradeoff, compare joint and alternative private release strategies, and examine the effect of network dimension on state clipping.