Latent Ornstein–Uhlenbeck-Driven Log-ARCH Models: Continuous-Time Stochastic Dynamics, Exact Discretization, and Long-Span QML Theory

We propose a latent Ornstein–Uhlenbeck (OU)-driven log-ARCH model in which a continuous-time stochastic factor governs the volatility level while a sparse lag structure captures volatility feedback. The OU process is discretized exactly, yielding a linear Gaussian state transition, while the log transformation produces a linear state-space representation with a non-Gaussian observation disturbance. We therefore employ a Gaussian quasi-likelihood based on the Kalman filter. A two-stage procedure is developed: an existence-based pilot screening rule first identifies the active lag set with probability approaching one under explicit pilot approximation and pseudo-true separation conditions, after which the finite-dimensional structural parameters are estimated by Gaussian quasi-maximum likelihood. Under a fixed sampling interval and long-span asymptotics, and under the stated identification, mixing, and score central limit conditions, we establish consistency and asymptotic normality of the structural estimator. For the latent state, we do not claim N−1/2 convergence; instead, the Gaussian quasi-smoother covariance converges to a non-degenerate steady-state limit. Monte Carlo experiments distinguish the theorem-aligned fixed-threshold screening rule from a practical sign-flip calibration device. An enlarged experiment with M=1000 replications documents substantial contraction of the median bias and interquartile range of the mean-reversion estimator as the sample size increases, although a residual upper tail remains at moderate sample sizes. The screening experiments show improving support recovery when the pseudo-true separation condition is favorable and deterioration in more difficult designs with weaker ARCH signals. Sensitivity analyses examine alternative maximal lag orders and kernels, and a standardized Student-t8 experiment is reported strictly as a misspecification diagnostic rather than as evidence for the Gaussian-theory statements.