Long-Memory and Features of Fluctuation in A Fractional Generalized Cauchy Process

Abstract

A fractional generalized Cauchy process (FGCP) is studied, which may give the same probability density function as the ordinary generalized Cauchy process. The exact solution of the Fokker-Planck equation for FGCP is given with the aid of the inverse Lévy transform. The associated eigenvalue problem is clarified. It is also exhibited the natures of long-memory, fractal, and volatility clustering associated with the FGCP.

Superstatistics and System Identification for a Class of Generalized Cauchy Processes

“…This formulation tells us that discrete and continuous eigenvalues are relevant [3], only from the observation of the potential function U (z) in Fig.1. Figure 2 shows the time series of y(t); further, the probability density P s (y) = …”

“…By a transformation of equation (15) into the Schrödinger type equation, one can see that discrete and continuous eigenvalues are relevant [3], only from the observation of the potential function U (y) in Fig.3:…”

“…Possible generalizations with a third order nonlinearity [2] and with a long-memory [3] are given. According to Beck-Cohen superstatistics [4], the model is derived by Bayes' theorem: P s (u) = ∫ P s (u|β)g(β) dβ , under the Gausssian distribution P s (u|β) with the Gamma distribution, g(β) = ν µ Γ(µ) β µ−1 exp(−νβ) .…”