1,605 research outputs found
Coupled continuous time random walks in finance
Continuous time random walks (CTRWs) are used in physics to model anomalous
diffusion, by incorporating a random waiting time between particle jumps. In
finance, the particle jumps are log-returns and the waiting times measure delay
between transactions. These two random variables (log-return and waiting time)
are typically not independent. For these coupled CTRW models, we can now
compute the limiting stochastic process (just like Brownian motion is the limit
of a simple random walk), even in the case of heavy tailed (power-law) price
jumps and/or waiting times. The probability density functions for this limit
process solve fractional partial differential equations. In some cases, these
equations can be explicitly solved to yield descriptions of long-term price
changes, based on a high-resolution model of individual trades that includes
the statistical dependence between waiting times and the subsequent
log-returns. In the heavy tailed case, this involves operator stable space-time
random vectors that generalize the familiar stable models. In this paper, we
will review the fundamental theory and present two applications with
tick-by-tick stock and futures data.Comment: 7 pages, 2 figures. Paper presented at the Econophysics Colloquium,
Canberra, Australia, November 200
Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy
Spatial non-locality of space-fractional viscoelastic equations of motion is
studied. Relaxation effects are accounted for by replacing second-order time
derivatives by lower-order fractional derivatives and their generalizations. It
is shown that space-fractional equations of motion of an order strictly less
than 2 allow for a new kind anisotropy, associated with angular dependence of
non-local interactions between stress and strain at different material points.
Constitutive equations of such viscoelastic media are determined. Explicit
fundamental solutions of the Cauchy problem are constructed for some cases
isotropic and anisotropic non-locality
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
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
The periodic repolarization dynamics index identifies changes in ventricular repolarization oscillations associated with music-induced emotions
The effect of music on cardiovascular dynamics may be useful in a variety of clinical settings. The aim of this study was to assess whether listening to music characterized by different emotional valence affected ventricular periodic repolarization dynamics (PRD), a recently-proposed non-invasive index of sympathetic ventricular modulation. The 12 lead ECG was recorded in 71 healthy volunteers exposed to six 90 s excerpts of pleasant music and unpleasant acoustic stimuli as well as six 90 s intervals of silence. A 20 s interval was allowed between excerpts during which the participants were asked to evaluate the previous excerpt. A simulation study was carried out to assess the capability of the algorithm of tracking fast small changes in PRD. The simulation study shows that the algorithm implemented in this study has a time-frequency resolution sufficient to capture the fast dynamics observed in this study. PRD were higher during listening to both pleasant and unpleasant music than during silence. There was a (weak) trend for the PRD to be higher during listening to pleasant than unpleasant music that may indicate the existence of a (weak) interaction between the valence of music-induced emotions and sympathetic ventricular response. The PRD significantly increased during the 20 s interval in between conditions, possibly reflecting a sympathetic response to the evaluation task and/or to the expectation of the following excerpt
Some Insights in Superdiffusive Transport
In this paper we deal with high-order corrections for the Fractional
Derivative approach to anomalous diffusion, in super-diffusive regime, which
become relevand whenever one attempts to describe the behavior of particles
close to normal diffusion.Comment: 14 pages, 7 figure
Monte Carlo evaluation of FADE approach to anomalous kinetics
In this paper we propose a comparison between the CTRW (Monte Carlo) and
Fractional Derivative approaches to the modelling of anomalous diffusion
phenomena in the presence of an advection field. Galilei variant and invariant
schemes are revised.Comment: 13 pages, 6 figure
Creep, Relaxation and Viscosity Properties for Basic Fractional Models in Rheology
The purpose of this paper is twofold: from one side we provide a general
survey to the viscoelastic models constructed via fractional calculus and from
the other side we intend to analyze the basic fractional models as far as their
creep, relaxation and viscosity properties are considered. The basic models are
those that generalize via derivatives of fractional order the classical
mechanical models characterized by two, three and four parameters, that we
refer to as Kelvin-Voigt, Maxwell, Zener, anti-Zener and Burgers. For each
fractional model we provide plots of the creep compliance, relaxation modulus
and effective viscosity in non dimensional form in terms of a suitable time
scale for different values of the order of fractional derivative. We also
discuss the role of the order of fractional derivative in modifying the
properties of the classical models.Comment: 41 pages, 8 figure
Mixtures of compound Poisson processes as models of tick-by-tick financial data
A model for the phenomenological description of tick-by-tick share prices in
a stock exchange is introduced. It is based on mixtures of compound Poisson
processes. Preliminary results based on Monte Carlo simulation show that this
model can reproduce various stylized facts.Comment: 12 pages, 6 figures, to appear in a special issue of Chaos, Solitons
and Fractal
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