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

TL;DR

This paper develops a discretization method for continuous time discrete scale invariant processes, introduces a spectral representation, and proposes a new, more accurate estimation method for the time-dependent Hurst parameter, validated through simulations and real data.

Contribution

It introduces a novel discretization approach, spectral characterization, and an improved estimation method for the Hurst parameter of continuous time DSI processes.

Findings

New discretization scheme for continuous time DSI processes

Spectral representation and spectrum estimation method

Enhanced accuracy in estimating time-dependent Hurst parameter

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.