927 research outputs found
A Poster about the Old History of Fractional Calculus
MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22The fractional calculus (FC) is an area of intensive research and development. In a previous paper and poster we tried to exhibit its recent state, surveying the period of 1966-2010. The poster accompanying the present note illustrates the major contributions during the period 1695-1970, the "old history" of FC
Mean Exit Time and Survival Probability within the CTRW Formalism
An intense research on financial market microstructure is presently in
progress. Continuous time random walks (CTRWs) are general models capable to
capture the small-scale properties that high frequency data series show. The
use of CTRW models in the analysis of financial problems is quite recent and
their potentials have not been fully developed. Here we present two (closely
related) applications of great interest in risk control. In the first place, we
will review the problem of modelling the behaviour of the mean exit time (MET)
of a process out of a given region of fixed size. The surveyed stochastic
processes are the cumulative returns of asset prices. The link between the
value of the MET and the timescale of the market fluctuations of a certain
degree is crystal clear. In this sense, MET value may help, for instance, in
deciding the optimal time horizon for the investment. The MET is, however, one
among the statistics of a distribution of bigger interest: the survival
probability (SP), the likelihood that after some lapse of time a process
remains inside the given region without having crossed its boundaries. The
final part of the article is devoted to the study of this quantity. Note that
the use of SPs may outperform the standard "Value at Risk" (VaR) method for two
reasons: we can consider other market dynamics than the limited Wiener process
and, even in this case, a risk level derived from the SP will ensure (within
the desired quintile) that the quoted value of the portfolio will not leave the
safety zone. We present some preliminary theoretical and applied results
concerning this topic.Comment: 10 pages, 2 figures, revtex4; corrected typos, to appear in the APFA5
proceeding
Recent history of fractional calculus
This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date
Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
A detailed study is presented for a large class of uncoupled continuous-time
random walks (CTRWs). The master equation is solved for the Mittag-Leffler
survival probability. The properly scaled diffusive limit of the master
equation is taken and its relation with the fractional diffusion equation is
discussed. Finally, some common objections found in the literature are
thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page
Activity autocorrelation in financial markets. A comparative study between several models
We study the activity, i.e., the number of transactions per unit time, of
financial markets. Using the diffusion entropy technique we show that the
autocorrelation of the activity is caused by the presence of peaks whose time
distances are distributed following an asymptotic power law which ultimately
recovers the Poissonian behavior. We discuss these results in comparison with
ARCH models, stochastic volatility models and multi-agent models showing that
ARCH and stochastic volatility models better describe the observed experimental
evidences.Comment: 15 pages, 4 figure
Fractional Fokker-Planck Equation for Ultraslow Kinetics
Several classes of physical systems exhibit ultraslow diffusion for which the
mean squared displacement at long times grows as a power of the logarithm of
time ("strong anomaly") and share the interesting property that the probability
distribution of particle's position at long times is a double-sided
exponential. We show that such behaviors can be adequately described by a
distributed-order fractional Fokker-Planck equations with a power-law
weighting-function. We discuss the equations and the properties of their
solutions, and connect this description with a scheme based on continuous-time
random walks
Truncation effects in superdiffusive front propagation with L\'evy flights
A numerical and analytical study of the role of exponentially truncated
L\'evy flights in the superdiffusive propagation of fronts in
reaction-diffusion systems is presented. The study is based on a variation of
the Fisher-Kolmogorov equation where the diffusion operator is replaced by a
-truncated fractional derivative of order where
is the characteristic truncation length scale. For there is no
truncation and fronts exhibit exponential acceleration and algebraic decaying
tails. It is shown that for this phenomenology prevails in the
intermediate asymptotic regime where
is the diffusion constant. Outside the intermediate asymptotic regime,
i.e. for , the tail of the front exhibits the tempered decay
, the acceleration is transient, and
the front velocity, , approaches the terminal speed as , where it is assumed that
with denoting the growth rate of the
reaction kinetics. However, the convergence of this process is algebraic, , which is very slow compared to the exponential
convergence observed in the diffusive (Gaussian) case. An over-truncated regime
in which the characteristic truncation length scale is shorter than the length
scale of the decay of the initial condition, , is also identified. In
this extreme regime, fronts exhibit exponential tails, ,
and move at the constant velocity, .Comment: Accepted for publication in Phys. Rev. E (Feb. 2009
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