Fractional derivatives of random walks: time series with long-time memory.

  title={Fractional derivatives of random walks: time series with long-time memory.},
  author={H. Eduardo Roman and Markus Porto},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  volume={78 3 Pt 1},
  • H. Roman, M. Porto
  • Published 19 June 2008
  • Mathematics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly decaying autocorrelation functions. The relation between these correlated walks and the well-known fractionally integrated autoregressive models with conditional heteroskedasticity (FIGARCH), commonly used in econometric studies, is discussed. The application of correlated… 
Estimating market index valuation from macroeconomic trends
We discuss USA stock market data from 1789 until 2020, focusing our attention on the S&P 500 index (1957–2020). We find that the data can be split into two periods, (1789–1948) and (1948–2020),
Entropy of timekeeping in a mechanical clock
The dynamics of an unique type of clock mechanism known as grasshopper escapement is investigated with the aim of evaluating its accuracy in a noisy environment. It is demonstrated that the clock’s
Disordered Random Walks
By treating steps of a random walk as components of a random vector, a random walk model is constructed resorting to the procedure used to generate disordered random matrices. It is shown that the
Synchronization of fractional chaotic complex networks with distributed delays
In this paper, we extend Lyapunov–Krasovskii stability theorem to fractional system with distributed delays. We take the integer derivatives instead of fractional derivatives to cope with the
Further properties of Osler's generalized fractional integrals and derivatives with respect to another function
In this paper we discuss fractional integrals and fractional derivatives of a function with respect to another function. We present some fundamental properties for both types of fractional operators,
Differentiable potentials and metallic states in disordered one-dimensional systems
We provide evidence that as a general rule Anderson localization effects become weaker as the degree of differentiability of the disordered potential increases. In one dimension a band of metallic
Localization in fractal and multifractal media
The propagation of waves in highly inhomogeneous media is a problem of interest in multiple fields including seismology, acoustics and electromagnetism. It is also relevant for technological
Fractional dynamics in DNA
Realization of a fractional-order RLC circuit via constant phase element
In the paper, a fractional-order RLC circuit is presented. The circuit is realized by using a fractional-order capacitor. This is realized by using carbon black dispersed in a polymeric matrix.


Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance
Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable
Introduction to Econophysics
This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena,
The Econometrics of Financial Markets
This book is an ambitious effort by three well-known and well-respected scholars to fill an acknowledged void in the literature—a text covering the burgeoning field of empirical finance. As the
Applications Of Fractional Calculus In Physics
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences,
however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)
  • Rev. Lett. 81, 729
  • 1998
  • Rev. E 63, 036128
  • 2001
Fractional differential equations
  • Phys. J. B 38, 671
  • 2004
Quantitative Finance 1
  • 237
  • 2001