Hurst estimation of scale invariant processes with drift and stationary increments

Abstract: The characteristic feature of the discrete scale invariant (DSI) processes is the invariance of their finite dimensional distributions by dilation for certain scaling factor. DSI process with piecewise linear drift and stationary increments inside prescribed scale intervals is introduced and studied. To identify the structure of the process, first we determine the scale intervals, their linear drifts and eliminate them. Then a new method for the estimation of the Hurst parameter of such DSI processes is presented and applied to some period of the Dow Jones indices. This method is based on fixed number equally spaced samples inside successive scale intervals. We also present some efficient method for estimating Hurst parameter of self-similar processes with stationary increments. We compare the performance of this method with the celebrated FA, DFA and DMA on the simulated data of fractional Brownian motion.

• The paper introduces a robust method for estimating the Hurst exponent in DSI processes by systematically removing piecewise linear drift.
• Flexible sampling and local regression within scale intervals enhance accuracy compared to traditional FA, DFA, and DMA methods.
• Validated on financial time series, the approach effectively transforms nonstationary data into stationary increments for reliable estimation.

Hurst Estimation in Discrete Scale Invariant Processes with Drift and Stationary Increments

Overview

The study systematically develops a new methodology for estimating the Hurst exponent in discrete scale invariant (DSI) processes that exhibit both stationary increments and piecewise linear drifts. The framework extends previous scale invariance analysis by incorporating drift elimination, robust parameter estimation, and comparison with standard approaches for long-memory processes. Real-world application to stock market data, notably the Dow Jones indices, is provided to substantiate the practical efficiency and validity of the proposed techniques.

Discrete Scale Invariant Processes with Drift Structure

Discrete scale invariance characterizes processes invariant under a discrete set of dilations. In applications such as financial time series, natural images, and network traffic, such scale invariance frequently appears as a local, rather than global, property, often accompanied by deterministic trends or drifts.

The authors formalize DSI processes with drift as comprising a pure DSI component and an additional, potentially piecewise, linear deterministic drift. In practice, this drift alters mean behavior across distinct scale intervals. The paper adopts regression within each detected scale interval to estimate and remove this drift, isolating the intrinsic scale-invariant dynamics for further analysis.

Flexible Sampling and Scale Interval Detection

An essential prerequisite for meaningful Hurst parameter estimation in DSI processes is the precise identification of scale intervals. The methodology employs parabolic fitting to delineate such intervals, consistent with observed regularity in many real-world scale-invariant phenomena. Within each detected scale interval, the sampling is made flexible by choosing a fixed number of equally spaced points, facilitating robust estimation despite varying interval lengths.

The approach allows for:

• Accurate adaptation to local DSI behavior,
• Systematic drift elimination per interval,
• Transformation of nonstationary raw series into stationary increment sequences by removing deterministic components.

Hurst Parameter Estimation Methodology

After drift removal, the main contribution is the estimation of the Hurst exponent $H$ based on variance scaling of first and second order increments across scale intervals. Given the stationary increment assumption within each scale interval, the sample variances of these increments are shown to relate across consecutive intervals by the scale parameter $\lambda$ and Hurst exponent through a logarithmic relationship.

Denoting $S_k^2$ as the sample variance in the $k^{th}$ interval, the estimator for $H$ in the first-order increment setting is:

$$\hat{H} = \frac{1}{2 \log \lambda} \log \left( \frac{S_k^2}{S_{k-1}^2} \right)$$

This is averaged over all available intervals for robustness. Second-order differencing and corresponding variance provide a refined estimator, particularly effective for higher values of $H$.

This scheme is benchmarked against standard FA, DFA, and DMA estimators in the context of fractional Brownian motion (fBm) with and without drift.

Application to Financial Time Series

Applying the methodology to the Dow Jones index over a multi-year span, the scale intervals and piecewise linear drifts are identified via regression and parabolic fitting. After drift removal, sample variances of increments are computed per interval, and Hurst exponents are estimated using the developed formula.

The results demonstrate:

• Estimated scale parameters across intervals in the range $1.18$–$1.80$,
• Interval-specific Hurst exponents $\approx 0.14$–$0.81$,
• Mean Hurst exponent estimates close to $0.5$ after local drift removal, corresponding to nearly Brownian behavior in the drift-corrected increments,
• Substantially reduced outlier estimate magnitudes compared to global drift removal or no drift correction, supporting the claim that local drift elimination is crucial for accurate Hurst estimation in nonstationary real-world series.

Comparative Performance Evaluation

Simulated experiments on fBm processes—both with and without drift—show that the proposed estimation approaches, particularly second-difference-based (diff2), consistently achieve lower mean squared errors (MSE) than FA, DFA, and DMA. Notably:

• The first-difference estimator is most effective for $H \in [0.1, 0.5]$,
• The second-difference estimator outperforms all others for $H \in [0.6, 0.9]$,
• For high-precision scenarios and large samples, the methodology provides tighter concentration around true $H$ values.

Theoretical and Practical Implications

The work addresses limitations of representative methods in the presence of deterministic trends and locally defined scale invariance. The introduction of flexible, per-interval drift correction and variance ratio-based Hurst estimation directly addresses known sources of bias in FA-DFA-DMA-like approaches.

Practically, this supports more accurate assessment of memory and roughness in financial markets and similar domains, specifically where trend and nonstationarity coexist with scale-invariant stochastic structure.

Theoretically, the methodology encourages further development of estimation procedures for nonstationary scale-invariant processes and motivates extensions to multidimensional, multivariate, or higher-order scale invariant systems with drift or other deterministic components.

Conclusion

This study introduces a statistically rigorous, robust, and empirically validated approach to Hurst exponent estimation in discrete scale-invariant processes with piecewise linear drift. By explicitly identifying and removing locally linear drifts, and by estimating $H$ via within-interval increment statistics, the methodology demonstrates significant improvement in estimation accuracy over widely used alternatives. The approach holds substantial promise for applications in econometrics, geophysics, and any context where scale invariance is masked by structural trends or drifts. Future research directions include extension to joint estimation of Hurst and drift parameters in more complex or multivariate DSI fields.