We provide a device, called the local predictor, which extends the idea of the predictable compensator. It is shown that a fBm with the Hurst index greater than 1/2 coincides with its local predictor while fBm with the Hurst index smaller than 1/2 does not admit any local predictor. Mathematics Subject Classification (2000). Primary 60G07; Secondary 60G15, 60G48, 60G25.

This paper is a condensed introduction to self-similarity, self-similar processes, and the estimation of the Hurst parameter in the context of time series analysis. It gives an overview of the literature on this subject and provides some assistance in implementing Hurst parameter estimators and carrying out experiments with empirical time series.

This paper presents a study of the Hurst index estimation in the case of fractional Ornstein–Uhlenbeck and geometric Brownian motion models. The performance of the estimators is studied both with respect to the value of the Hurst index and the length of sample paths.

The Hurst parameter captures the amount of long-range dependence (LRD) in a time series. There are several methods to estimate the Hurst parameter, being the most popular: the variance-time plot, the R/S plot, the periodogram, and Whittle’s estimator. The first three are graphical methods, and the estimation accuracy depends on how the plot is interpreted and calculated. In contrast, Whittle’s ...

This research studies three mathematical models, namely geometric Brownian motion (GBM), fractional (GFBM) model which was developed by adding the Hurst parameter to GBM characterize long-memory phenomenon, and Merton jump-diffusion (MJD) captures shocks via GBM. study sets out forecast Malaysia rubber prices for six months period beginning in January 2022 ending June 2022, involves four main s...

This paper is a condensed introduction to self-similarity, self-similar processes, and the estimation of the Hurst parameter in the context of time series analysis. It gives an overview of the literature on this subject and provides some assistance in implementing Hurst parameter estimators and carrying out experiments with empirical time series.

We consider a passive scalar in a periodic shear flow perturbed by an additive fractional noise with the Hurst exponent H ∈ (0, 1). We establish a diffusive homogenization limit for the tracer when the Hurst exponent H ∈ (0, 1/2). We also identify an intermediate range of times when the tracer behaves diffusively even when H ∈ (1/2, 1). The proof is based on an auxiliary limit theorem for an ad...