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

Paper: arXiv 2603.05119

Authors: Sourojyoti Barick

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework combining minimum density power divergence estimators (MDPDE) with extreme value theory (Gumbel limits) for jump detection in high-frequency CKLS/CIR models. While the theoretical derivations and classification consistency proofs are extensive, empirical validation relies heavily on simulations rather than extensive real-world high-frequency financial datasets.

Research Flowchart

  flowchart TD
    A["<b>Research Goal:</b> Robust jump detection in high-frequency CKLS jump-diffusion models"] --> B["<b>Input Data:</b> Discretely observed high-frequency time series (Δ<sub>n</sub> → 0)"]
    
    B --> C["<b>Step 1: MDPDE Parameter Estimation</b><br/>Robust estimation of drift and diffusion parameters"]
    
    C --> D["<b>Step 2: Residual Normalization & Scale Separation</b><br/>Exploit scale gap: Diffusion ~ O(√Δ<sub>n</sub>) vs. Jumps ~ O(1)"]
    
    D --> E["<b>Step 3: Gumbel Thresholding</b><br/>Apply EVT threshold based on Gumbel limit distribution"]
    
    E --> F["<b>Key Outcomes:</b><br/>1. Classification consistency (P → 1)<br/>2. Reduced spurious jump detections"]

Cited References (47)

1. Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets (2016) — link
2. An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process (2012) — link
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4. A Theory of the Term Structure of Interest Rates (1985)
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6. Minimum Density Power Divergence Estimator for Diffusion Processes (2013) — link
7. Estimation of an Ergodic Diffusion from Discrete Observations (1997)
8. Adaptive estimation of an ergodic diffusion process based on sampled data (2012)

0 of 47 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.