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

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 $\sqrt{Δ_n}$, 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.

💡 Research Summary

The paper develops a robust, parametric framework for detecting jumps in high‑frequency observations of Cox‑Ingersoll‑Ross (CIR) and the more general Chan‑Karolyi‑Longstaff‑Sanders (CKLS) diffusion models. The authors start by recalling that, under the “infill” asymptotic regime (Δₙ → 0, nΔₙ → ∞), diffusion increments shrink at the rate √Δₙ while jump increments remain of order one. Classical estimators such as maximum‑likelihood (MLE) or conditional least squares (CLS) treat all increments equally and therefore become highly sensitive to the occasional large outliers generated by jumps, leading to bias and loss of efficiency.

To mitigate this problem, the paper adopts the Minimum Density Power Divergence Estimator (MDPDE) introduced by Basu et al. (1998). The MDPDE minimizes a power‑divergence objective (H_n^{(α)}(θ)) that interpolates between the log‑likelihood (α = 0) and a highly robust criterion (α > 0). The tuning parameter α controls the trade‑off: larger α down‑weights observations that have low model‑implied density, i.e., potential jumps, while preserving consistency under the correctly specified diffusion model. Under standard regularity conditions and the high‑frequency scheme, the authors prove that the MDPDE is consistent and asymptotically normal (Result 3), with a bounded influence function that guarantees robustness against extreme increments.

Using the robust estimates (\hat θ(α)), the authors construct standardized residuals
(R_i = \frac{ΔX_i - \hat μ_i}{\hat σ \sqrt{Δ_n}}).
In pure diffusion intervals these residuals behave like standard normal variables, whereas in intervals containing a jump they exhibit large absolute values. The key theoretical contribution is the extreme‑value analysis of the maximum absolute residual (M_n = \max_i |R_i|). The authors show that, over diffusion intervals, (M_n) converges in distribution to the Gumbel extreme‑value law. This result yields an explicit, asymptotically valid detection threshold (c_{n,γ}) that depends only on the sampling design and the chosen significance level.

A jump‑detection rule is then defined: declare a jump whenever (M_n > c_{n,γ}). The authors prove “classification consistency”: as n → ∞ and Δₙ → 0, the probability of correctly labeling every jump and every diffusion increment converges to one. This establishes that the procedure not only controls false positives asymptotically but also achieves perfect power in the limit.

Monte‑Carlo experiments are conducted for the CIR case (γ = ½) and several CKLS specifications (γ = 0.6, 0.8). Different jump intensities, jump size distributions, and α‑values (0, 0.1, 0.2, 0.3) are examined. The simulations confirm that the MDPDE‑based method dramatically reduces spurious detections compared with the classical MLE‑based test, while retaining comparable detection power. An α around 0.2 provides a favorable balance between robustness and efficiency. The Gumbel‑based threshold remains accurate even when jumps follow an infinite‑activity Lévy process, demonstrating the method’s flexibility beyond finite‑activity settings.

In conclusion, the paper offers a theoretically rigorous and practically robust approach to jump identification that fully exploits the parametric structure of CIR/CKLS models. By integrating divergence‑based robust estimation with extreme‑value theory, the authors achieve a unified inference procedure that is both statistically efficient under pure diffusion and resilient to contamination by jumps. The methodology is directly applicable to high‑frequency interest‑rate modeling, option pricing, and risk management, and opens avenues for extensions to multivariate settings, irregular sampling schemes, and real‑time online detection algorithms.