166 research outputs found
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
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
Theory of Second and Higher Order Stochastic Processes
This paper presents a general approach to linear stochastic processes driven
by various random noises. Mathematically, such processes are described by
linear stochastic differential equations of arbitrary order (the simplest
non-trivial example is , where is not a Gaussian white
noise). The stochastic process is discretized into time-steps, all possible
realizations are summed up and the continuum limit is taken. This procedure
often yields closed form formulas for the joint probability distributions.
Completely worked out examples include all Gaussian random forces and a large
class of Markovian (non-Gaussian) forces. This approach is also useful for
deriving Fokker-Planck equations for the probability distribution functions.
This is worked out for Gaussian noises and for the Markovian dichotomous noise.Comment: 35 pages, PlainTex, accepted for publication in Phys Rev. E
Integrated random processes exhibiting long tails, finite moments and 1/f spectra
A dynamical model based on a continuous addition of colored shot noises is
presented. The resulting process is colored and non-Gaussian. A general
expression for the characteristic function of the process is obtained, which,
after a scaling assumption, takes on a form that is the basis of the results
derived in the rest of the paper. One of these is an expansion for the
cumulants, which are all finite, subject to mild conditions on the functions
defining the process. This is in contrast with the Levy distribution -which can
be obtained from our model in certain limits- which has no finite moments. The
evaluation of the power spectrum and the form of the probability density
function in the tails of the distribution shows that the model exhibits a 1/f
spectrum and long tails in a natural way. A careful analysis of the
characteristic function shows that it may be separated into a part representing
a Levy processes together with another part representing the deviation of our
model from the Levy process. This allows our process to be viewed as a
generalization of the Levy process which has finite moments.Comment: Revtex (aps), 15 pages, no figures. Submitted to Phys. Rev.
Extreme times in financial markets
We apply the theory of continuous time random walks to study some aspects of
the extreme value problem applied to financial time series. We focus our
attention on extreme times, specifically the mean exit time and the mean
first-passage time. We set the general equations for these extremes and
evaluate the mean exit time for actual data.Comment: 6 pages, 3 figure
Heat Conduction in One-Dimensional chain of Hard Discs with Substrate Potential
Heat conduction of one-dimensional chain of equivalent rigid particles in the
field of external on-site potential is considered. Zero diameters of the
particles correspond to exactly integrable case with divergent heat conduction
coefficient. By means of simple analytical model it is demonstrated that for
any nonzero particle size the integrability is violated and the heat conduction
coefficient converges. The result of the analytical computation is verified by
means of numerical simulation in a plausible diapason of parameters and good
agreement is observedComment: 14 pages, 7 figure
A continuous time random walk model for financial distributions
We apply the formalism of the continuous time random walk to the study of
financial data. The entire distribution of prices can be obtained once two
auxiliary densities are known. These are the probability densities for the
pausing time between successive jumps and the corresponding probability density
for the magnitude of a jump. We have applied the formalism to data on the US
dollar/Deutsche Mark future exchange, finding good agreement between theory and
the observed data.Comment: 14 pages, 5 figures, revtex4, submitted for publicatio
Coherence resonance in a network of FitzHugh-Nagumo systems: interplay of noise, time-delay and topology
We systematically investigate the phenomena of coherence resonance in
time-delay coupled networks of FitzHugh-Nagumo elements in the excitable
regime. Using numerical simulations, we examine the interplay of noise,
time-delayed coupling and network topology in the generation of coherence
resonance. In the deterministic case, we show that the delay-induced dynamics
is independent of the number of nearest neighbors and the system size. In the
presence of noise, we demonstrate the possibility of controlling coherence
resonance by varying the time-delay and the number of nearest neighbors. For a
locally coupled ring, we show that the time-delay weakens coherence resonance.
For nonlocal coupling with appropriate time-delays, both enhancement and
weakening of coherence resonance are possible
Functional characterization of generalized Langevin equations
We present an exact functional formalism to deal with linear Langevin
equations with arbitrary memory kernels and driven by any noise structure
characterized through its characteristic functional. No others hypothesis are
assumed over the noise, neither the fluctuation dissipation theorem. We found
that the characteristic functional of the linear process can be expressed in
terms of noise's functional and the Green function of the deterministic
(memory-like) dissipative dynamics. This object allow us to get a procedure to
calculate all the Kolmogorov hierarchy of the non-Markov process. As examples
we have characterized through the 1-time probability a noise-induced interplay
between the dissipative dynamics and the structure of different noises.
Conditions that lead to non-Gaussian statistics and distributions with long
tails are analyzed. The introduction of arbitrary fluctuations in fractional
Langevin equations have also been pointed out
Sistemes socioeconòmics i financers
Els mercats fi nancers, entre molts altres contextos socials i econòmics, amaguen diverses relacions amb la �� sica estadís�� ca.
Sense anar més lluny, el model matemà�� c de les co�� tzacions fi nanceres és el mateix u�� litzat per a la teoria de gasos o per
les par�� cules en suspensió en un líquid. En aquest ar�� cle recorrem la trajectòria de l'anomenada econo�� sica des de 1900 i
presentem algunes de les contribucions a la matèria feta per membres de Complexitat.CAT
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