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Testing for jumps in processes with integral fractional part and jump-robust inference on the Hurst exponent

Markus Bibinger, Michael Sonntag

TL;DR

The paper tackles detecting jumps in discretely observed integral fractional processes with unknown $H\in(0,1)$ by adapting a max-type test based on suitably normalized second-order increments. It develops a jump-robust inference framework for the Hurst exponent using small-power second-order variations, establishing a standard Gumbel limit under no jumps and consistency under jumps, along with a fast $O_p(n^{-1})$ jump localization rate. The results demonstrate robustness of H-estimation to jumps for rough volatility and provide a sequential procedure to localize multiple jumps, supported by simulations and an empirical volatility data application. Practically, the methods enable reliable jump detection and filtering in rough volatility settings, improving inference and forecasting in financial time series where jumps and roughness interact.

Abstract

We develop and investigate a test for jumps based on high-frequency observations of a fractional process with an additive jump component. The Hurst exponent of the fractional process is unknown. The asymptotic theory under infill asymptotics builds upon extreme value theory for weakly dependent, stationary time series and extends techniques for the semimartingale case from the literature. It is shown that the statistic on which the test is based on weakly converges to a Gumbel distribution under the null hypothesis of no jumps. We prove consistency under the alternative hypothesis when there are jumps. Moreover, we establish convergence rates for local alternatives and consistent estimation of jump times. In the process, we show that inference on the Hurst exponent of a rough fractional process is robust with respect to jumps. This provides an important insight for the growing literature on rough volatility. We demonstrate sound finite-sample properties in a simulation study and showcase the applicability of our methods in an empirical example with a time series of volatilities.

Testing for jumps in processes with integral fractional part and jump-robust inference on the Hurst exponent

TL;DR

The paper tackles detecting jumps in discretely observed integral fractional processes with unknown by adapting a max-type test based on suitably normalized second-order increments. It develops a jump-robust inference framework for the Hurst exponent using small-power second-order variations, establishing a standard Gumbel limit under no jumps and consistency under jumps, along with a fast jump localization rate. The results demonstrate robustness of H-estimation to jumps for rough volatility and provide a sequential procedure to localize multiple jumps, supported by simulations and an empirical volatility data application. Practically, the methods enable reliable jump detection and filtering in rough volatility settings, improving inference and forecasting in financial time series where jumps and roughness interact.

Abstract

We develop and investigate a test for jumps based on high-frequency observations of a fractional process with an additive jump component. The Hurst exponent of the fractional process is unknown. The asymptotic theory under infill asymptotics builds upon extreme value theory for weakly dependent, stationary time series and extends techniques for the semimartingale case from the literature. It is shown that the statistic on which the test is based on weakly converges to a Gumbel distribution under the null hypothesis of no jumps. We prove consistency under the alternative hypothesis when there are jumps. Moreover, we establish convergence rates for local alternatives and consistent estimation of jump times. In the process, we show that inference on the Hurst exponent of a rough fractional process is robust with respect to jumps. This provides an important insight for the growing literature on rough volatility. We demonstrate sound finite-sample properties in a simulation study and showcase the applicability of our methods in an empirical example with a time series of volatilities.
Paper Structure (20 sections, 15 theorems, 100 equations, 6 figures, 2 tables)

This paper contains 20 sections, 15 theorems, 100 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model, assumptions, and testing problem
  3. Observation model
  4. Testing problem
  5. Regularity assumptions
  6. Construction of the tests
  7. Jump-robust inference on rough processes
  8. Asymptotic properties of the tests
  9. Asymptotic distribution under the null hypothesis
  10. Consistency and rate of convergence under the alternative
  11. Localization of jumps
  12. Simulations
  13. Data application
  14. Conclusion
  15. Proofs
  16. ...and 5 more sections

Key Result

Proposition 4.1

If $H\in(0,1/2)$, for any jump process $(J_t)$ with finite quadratic variation which is independent of $(Y_t)$, it holds under Assumption Annahmen that In particular, this implies that $n^{2H-1}\sum_{j=1}^n(\Delta_{n,j}^{(1)}X)^2\stackrel{\mathbb{P}}{\rightarrow}\int_0^1\sigma_s^2\,\mathrm{d} s$, as $n\to\infty$.

Figures (6)

  • Figure 1: Sample paths (top) and increments (bottom) with $H=0{.3}$ (left) and $H=0{.7}$ (right), and three jumps added which are highlighted by red circles and points.
  • Figure 2: Empirical distributions of test statistic under the null hypothesis, compared to the standard Gumbel limit distribution drawn with the black line, and under the alternative hypothesis with fix jump size $\Delta J_{\tau}=0{.}8$.
  • Figure 3: Empirical power against level $\gamma$ of jump sizes $\sqrt{2\log(n)}n^{-\gamma}$.
  • Figure 4: Empirical robustness in estimating the Hurst exponent.
  • Figure 5: Example time series of daily volatilities (left) and logarithmic daily volatilities (right).
  • ...and 1 more figures

Theorems & Definitions (21)

  • Proposition 4.1
  • Proposition 4.2
  • Theorem 1
  • Corollary 5.1
  • Theorem 2
  • Corollary 5.2
  • Proposition 5.3
  • Lemma 9.1
  • Lemma 9.2: Rescaled Minima
  • proof
  • ...and 11 more