Hurst Exponents and Delampertized Fractional Brownian Motions

Abstract

The inverse Lamperti transform of a fractional Brownian motion (fBm) is a stationary process. We determine the empirical Hurst exponent of such a composite process with the help of a regression of the log absolute moments of its increments, at various scales, on the corresponding log scales. This perceived Hurst exponent underestimates the Hurst exponent of the underlying fBm. We thus encounter some time series having a perceived Hurst exponent lower than [Formula: see text], but an underlying Hurst exponent higher than [Formula: see text]. This paves the way for short- and medium-term forecasting. Indeed, in such series, mean reversion predominates at high scales, whereas persistence is overriding at lower scales. We propose a way to characterize the Hurst horizon, namely a limit scale between these opposite behaviors. We show that the delampertized fBm, which mixes persistence and mean reversion, is relevant for financial time series, in particular for high-frequency foreign exchange rates. In our sample, the empirical Hurst horizon is always above 1[Formula: see text]h and 23[Formula: see text]min.

Long versus short time scales: the rough dilemma and beyond

“…In fact, this bias of the absolute-moment estimator in case of stationary time series is well documented e.g. in [Garcin, 2019]. The stationarity indeed tends to flatten the log-plot for high scales as emphasized in the previous subsection: the log-plot is thus nonlinear and the linear regression irrelevant.…”

“…However, when dealing with scales longer than two months, we should expect a deviation from the linear behaviour in the log-log plot for stationarized fBm processes (for example a fOU or the inverse Lamperti transform of a fBm, see e.g. [Cheridito et al, 2003, Garcin, 2019, Šapina et al, 2017). We illustrate the phenomenon in the simple case where k = 2, namely the second absolute moment for the fBm, that can be rewritten as follows:…”

The Hurst Exponent of Heart Rate Variability in Neonatal Stress, Based on a Mean-Reverting Fractional Lévy Stable Motion

“…Multifractal dynamics are thus more accurate in this framework [25,33]. But the fact that fBm is not well suited to HRV can be explained by many other model specifications than the sole multifractality of the process, like a time-dependent Hurst exponent [18] or a Lamperti transform of a fBm [19], which generalizes the meanreverting Ornstein-Uhlenbeck process. A rapid look at some healthy and at rest HRV time series shows besides a mean reversion [12,24].…”

Forecasting Value-at-Risk in turbulent stock markets via the local regularity of the price process

“…the exchange rates. These display some peculiarities; for example, the rates series are generally reputed stationary and the classical Brownian motion model is there transformed by the Ornstein-Uhlenbeck approach or the Lamperti transform; samely, fractional Brownian motion is made stationary using similar techniques (see, e.g., Cheridito Kawaguchi and Maejima (2003), Chronopoulou and Viens (2012), Flandrin et al (2003), Garcin (2019)). In addition to these standard approaches, it is worthwhile to quote the recent work by Garcin (2020), who -stressing the analogue with the evolution of the volatility models and by using a Fisher-like transformation -defines the Multifractional Process with Fractional Exponent (MPFE).…”