Discretization of Continuous Time Discrete Scale Invariant Processes: Estimation and Spectra

Abstract: Imposing some flexible sampling scheme we provide some discretization of continuous time discrete scale invariant (DSI) processes which is a subsidiary discrete time DSI process. Then by introducing some simple random measure we provide a second continuous time DSI process which provides a proper approximation of the first one. This enables us to provide a bilateral relation between covariance functions of the subsidiary process and the new continuous time processes. The time varying spectral representation of such continuous time DSI process is characterized, and its spectrum is estimated. Also, a new method for estimation time dependent Hurst parameter of such processes is provided which gives a more accurate estimation. The performance of this estimation method is studied via simulation. Finally this method is applied to the real data of S$&$P500 and Dow Jones indices for some special periods.

• The paper introduces a discretization scheme that transforms continuous time DSI processes into a subsidiary discrete representation using structured time partitions.
• It derives explicit covariance structures and a time-local spectral representation, linking DSI characteristics to periodic correlations for improved analysis.
• A novel piecewise Hurst estimator is proposed, significantly reducing mean squared error compared to global methods in financial time series.

Discretization, Estimation, and Spectral Analysis of Continuous Time Discrete Scale Invariant Processes

Introduction and Problem Statement

This work addresses the discretization and analysis of continuous time discrete scale invariant (DSI) processes, providing new methodologies for covariance characterization, spectral representation, and estimation of non-constant Hurst parameters. While DSI processes exhibit invariance under dilations by specific scaling factors, many real-world data manifest non-stationary scaling behavior where the self-similarity exponent (the Hurst parameter $H$) is time-dependent rather than constant. Accurately analyzing such processes requires both valid discretization schemes and robust estimation techniques, especially in applications ranging from turbulence and geophysics to finance.

Discretization of Continuous Time DSI Processes

The paper establishes a flexible sampling scheme that transforms a continuous time DSI process into a subsidiary discrete time DSI process. Given a process $X(t)$ invariant under the scaling $t \mapsto \lambda t$, with non-constant $H(t)$, a partitioning of the positive real line into scale intervals and subintervals forms the foundation for discretization. Sampling at arbitrary points within the canonical scale interval is generalized across scale intervals via multiplicative group structure.

To link discrete and continuous representations, the authors define random measures on each subinterval, where the measure associated with subinterval $B_{j,i}$ aggregates increments of $X(t)$. This construction enables the explicit computation of covariance functions for the discretized process, which, in turn, allow for the definition of a new continuous time DSI process approximating the original, but now in terms of simple random measures and sample-based increments. The bilateral relationship between the covariance functions of the discrete and continuous processes is rigorously formalized, enabling further analytic and inferential development.

Covariance Structure and Spectral Representation

Building on the discretized framework, the paper derives detailed covariance structures for the continuous time DSI process expressed in terms of partitions, sampling schemes, and the DSI property. The covariance between process values at arbitrary points is explicitly related to the covariance structure of the discrete process increments, extended multiplicatively through the DSI scaling.

The authors further provide a time-varying spectral representation for the continuous time DSI process, leveraging the unitary operator spectral theory for periodically correlated (PC) sequences and the Lamperti transformation, which connects DSI and PC classes. The spectral density and representation are thus obtained by mapping from the discrete subsidiary process and using appropriate transformation results.

This synthesis bridges classical spectral analysis on Hilbert spaces with DSI processes, offering a tractable representation for both simulation and empirical data analysis.

Estimation of the Time-Dependent Hurst Parameter

A key contribution is a novel estimation procedure for the local Hurst parameter in DSI processes with piecewise approximate constancy in $H(t)$. The method partitions each scale interval into fixed-ratio subintervals and estimates a vector of Hurst indices, one for each subinterval, by computing quadratic variations and their ratios across consecutive scale intervals. The estimation procedure does not assume global self-similarity but respects local scaling features, which aligns with observed empirical behaviors in several domains.

Numerical simulation results indicate that this estimator achieves significantly lower mean squared error than the earlier method of Modarresi and Rezakhah ([16]), particularly when $H$ varies appreciably over time. For synthetic processes with four scale intervals and four subintervals each, the subinterval-based estimator substantially improves accuracy across a range of underlying $H$ values.

Empirical Results and Applications

The methodology is applied to financial time series: S&P500 and Dow Jones daily indices. Through the sampling and partitioning procedures, periods with evident DSI behavior are identified, and Hurst parameters are estimated in a piecewise manner. For the S&P500, two primary subinterval estimations are obtained: $H \approx 0.23$ for one segment and $H \approx 0.06$ for another, in contrast with a single global estimate $H = 0.16$ by the prior method. For Dow Jones, subinterval estimates of $H = [0.46, 0.56, 0.63, 0.50]$ contrast with a global estimate $H = 0.53$. These results highlight the estimator's ability to capture local scaling properties, with practical implications for modeling and risk estimation in financial markets.

Theoretical and Practical Implications

The paper provides a rigorous framework for analyzing stochastic processes with discrete scale invariance in continuous time, bridging gaps between discrete and continuous representations through structured discretization and random measure theory. The piecewise estimation method for Hurst parameters accommodates local variation, a property often observed in empirical data but overlooked by global models.

The formalization of time-localized spectral analysis in DSI processes has immediate consequences for statistical inference in applied domains, supporting more accurate modeling of real-world phenomena characterized by complex, non-stationary scaling—particularly in finance, geoscience, and network traffic modeling.

Conclusion

This work offers a comprehensive approach to the discretization, covariance, and spectral analysis of continuous time DSI processes, addressing both the mathematical structure and applied estimation challenges. The proposed subinterval-based Hurst estimator outperforms previous global estimators in terms of mean squared error, as demonstrated both in simulation and on financial time series. The analytical framework not only advances understanding of DSI processes with time-dependent scaling, but also sets the stage for further developments in multiscale modeling, locally adaptive inference, and the extension of these methods to other complex systems exhibiting discrete scale invariance.

Reference:

"Discretization of Continuous Time Discrete Scale Invariant Processes: Estimation and Spectra" (1601.04405)