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 $\sqrt{Δ_n}$, 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.

Figures (4)

• Figure 1: Effect of robustness on pointwise influence and likelihood contributions under jump contamination. Increasing $\alpha$ progressively bounds the influence of large, jump-induced increments while preserving the contribution of diffusion-driven observations.
• Figure 2: Visual comparison of true and detected jumps in the increment process $\Delta_i^n X$ for different values of the robustness parameter $\alpha$. denotes true jumps, whereas denotes jumps detected by the proposed procedure.
• Figure 3: F1 score as a function of the robustness parameter $\alpha$ for different sample sizes ($n$), jump intensities ($\lambda$), and jump mean values ($\mu_J$). Panels are arranged by sample size (rows) and jump mean (columns), while curves correspond to different jump intensities. The figure illustrates how the robustness parameter affects jump detection accuracy and how this effect varies with jump magnitude, jump frequency, and sampling resolution.
• Figure 4: Error metric ($d_M$) as a function of the robustness parameter $\alpha$ under varying sample sizes ($n$), jump intensities ($\lambda$), and jump mean values ($\mu_J$). Panels are arranged by sample size (rows) and jump mean (columns), while curves correspond to different jump intensities. The figure illustrates how the robustness tuning parameter influences the accuracy of jump parameter estimation and how this effect interacts with jump magnitude, jump frequency, and sampling resolution.

Theorems & Definitions (8)

• Remark 1
• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3: Consistency of parametric jump detection
• proof : Proof of the theorem \ref{['thm:parametric_LM']}
• proof : Proof of Theorem \ref{['thm:max_LM']}
• proof : Proof of Theorem \ref{['thm:robust_detection_consistency']}