Asymptotic Separability of Diffusion and Jump Components in High-Frequency CIR and CKLS Models

Abstract

This paper develops a robust parametric framework for jump detection in discretely observed CKLS-type jump-diffusion processes with high-frequency asymptotics, based on the minimum density power divergence estimator (MDPDE). The methodology exploits the intrinsic asymptotic scale separation between diffusion increments, which decay at rate $\sqrt{\Delta_n}$, and jump increments, which remain of non-vanishing stochastic magnitude. Using robust MDPDE-based estimators of the drift and diffusion coefficients, we construct standardized residuals whose extremal behavior provides a principled basis for statistical discrimination between continuous and discontinuous components. We establish that, over diffusion intervals, the maximum of the normalized residuals converges to the Gumbel extreme-value distribution, yielding an explicit and asymptotically valid detection threshold. Building on this result, we prove classification consistency of the proposed robust detection procedure: the probability of correctly identifying all jump and diffusion increments converges to one under proper asymptotics. The MDPDE-based normalization attenuates the influence of atypical increments and stabilizes the detection boundary in the presence of discontinuities. Simulation results confirm that robustness improves finite-sample stability and reduces spurious detections without compromising asymptotic validity. The proposed methodology provides a theoretically rigorous and practically resilient robust approach to jump identification in high-frequency stochastic systems.