Optimal Fusion of Delayed Price Feeds: Filtering, Identification, Model Choice and Surveillance

The information value of a price feed depends on its delay, noise persistence, and other feeds' histories. We study estimation, identification and surveillance for a Gaussian random walk observed through independent AR(1) feed noises with common persistence and integer delays. The exact steady-state filtering error for finitely many feeds reduces to scalar fractional-linear maps. Closed forms make covariance evaluation depend on the count of distinct ages rather than their magnitude; a recursion recovers the conditional mean. Every additional independent feed with finite delay has strictly positive information value, and increasing an individual feed's delay or noise variance strictly raises optimal mean-square error. For two feeds, we derive the stale stream's exact value and exponential decay with excess delay. An expansion near unit persistence gives a sufficient condition for an interior error maximum. Joint delay–persistence limits do not commute: streams can remain informative at large delays with sufficiently persistent noise. Gaussian quadratic-form concentration yields a uniform finite-sample bound for cross-covariance estimates and a condition for exact relative-delay recovery. Under compatible unanchored initial laws, common delay shifts preserve the observation law while changing current-price uncertainty; unbounded shifts preclude finite-width intervals with positive uniform coverage, whereas a bounded age bracket yields a conditional minimax risk bound. On a comparison graph, changes affecting at most s feeds are uniformly identifiable if and only if every connected component contains more than 2s vertices. Majority centering supplies stable recovery, and the concentration bound gives a finite-sample attribution guarantee across homogeneous segments. We formulate model comparison on a common observation space, distinguish exact Gaussian filtering from the interacting multiple model approximation, and construct an anytime-valid likelihood surveillance rule over a declared null family. Proofs specify initialization, support and information assumptions required by each guarantee. Independent scalar and augmented-state implementations, randomized configurations and synthetic experiments check the formulas.