Lévy Processes: Characterizing Volcanic and Financial Time Series

In this work, we use the Diffusion Entropy Analysis (DEA) to analyze and detect the scaling properties of time series from both emerging and well established markets as well as volcanic eruptions recorded by a seismic station, both financial and volcanic time series data are known to have high frequencies (i.e they are collected at an extremely fine scale). The objective is to determine the characterization i.e whether they follow a Gaussian or Lévy distribution. If they do follow a Lévy distribution we are then interested in finding if they are characterized by a Lévy walk which has a finite second moment or a Lévy flight which has an infinite second moment. We also seek to establish the existence of long- range correlations in these time series.That is we seek to determine if both time series are persistent (i.e have long-range correlation), anti-persistent or random. The results obtained from the DEA technique are compared with the Hurst R/S analysis and Detrended Fluctuation Analysis (DFA) methodologies. We conclude that given the scaling exponents δ derived from the DEA and H, α derived from the Hurst R/S analysis and DFA respectively, if 0.5 \u3c H, α, δ \u3c 1 the time series is said to exhibit long-range correlations and if 0 \u3c H, α, δ \u3c 0.5 the time series is said to be anti-persistent. Also for characterization, if δ is related to H or α by the relation δ =1/(3 − 2(H, α)), the time series is characterized by a Lévy walk. If δ = (H, α), the time series may be characterized by Fractional Brownian Motion (FBM) (i.e the time series is random), and finally if δ 6 = (H, α), the time series cannot be characterized by an FBM and this implies that the time series has an infinite second moment and is thus characterized by a Lévy flight