201 research outputs found
Single integro-differential wave equation for L\'evy walk
The integro-differential wave equation for the probability density function
for a classical one-dimensional L\'evy walk with continuous sample paths has
been derived. This equation involves a classical wave operator together with
memory integrals describing the spatio-temporal coupling of the L\'evy walk. It
is valid for any running time PDF and it does not involve any long-time
large-scale approximations. It generalizes the well-known telegraph equation
obtained from the persistent random walk. Several non-Markovian cases are
considered when the particle's velocity alternates at the gamma and power-law
distributed random times.Comment: 5 page
Nonlinear subdiffusive fractional equations and aggregation phenomenon
In this article we address the problem of the nonlinear interaction of
subdiffusive particles. We introduce the random walk model in which statistical
characteristics of a random walker such as escape rate and jump distribution
depend on the mean field density of particles. We derive a set of nonlinear
subdiffusive fractional master equations and consider their diffusion
approximations. We show that these equations describe the transition from an
intermediate subdiffusive regime to asymptotically normal advection-diffusion
transport regime. This transition is governed by nonlinear tempering parameter
that generalizes the standard linear tempering. We illustrate the general
results through the use of the examples from cell and population biology. We
find that a nonuniform anomalous exponent has a strong influence on the
aggregation phenomenon.Comment: 10 page
Sub-diffusion in External Potential: Anomalous hiding behind Normal
We propose a model of sub-diffusion in which an external force is acting on a
particle at all times not only at the moment of jump. The implication of this
assumption is the dependence of the random trapping time on the force with the
dramatic change of particles behavior compared to the standard continuous time
random walk model. Constant force leads to the transition from non-ergodic
sub-diffusion to seemingly ergodic diffusive behavior. However, we show it
remains anomalous in a sense that the diffusion coefficient depends on the
force and the anomalous exponent. For the quadratic potential we find that the
anomalous exponent defines not only the speed of convergence but also the
stationary distribution which is different from standard Boltzmann equilibrium.Comment: 6 pages, 3 figure
Stochastic arbitrage return and its implications for option pricing
The purpose of this work is to explore the role that arbitrage opportunities
play in pricing financial derivatives. We use a non-equilibrium model to set up
a stochastic portfolio, and for the random arbitrage return, we choose a
stationary ergodic random process rapidly varying in time. We exploit the fact
that option price and random arbitrage returns change on different time scales
which allows us to develop an asymptotic pricing theory involving the central
limit theorem for random processes. We restrict ourselves to finding pricing
bands for options rather than exact prices. The resulting pricing bands are
shown to be independent of the detailed statistical characteristics of the
arbitrage return. We find that the volatility "smile" can also be explained in
terms of random arbitrage opportunities.Comment: 14 pages, 3 fiqure
Asymptotic behavior of the solution of the space dependent variable order fractional diffusion equation: ultra-slow anomalous aggregation
We find for the first time the asymptotic representation of the solution to
the space dependent variable order fractional diffusion and Fokker-Planck
equations. We identify a new advection term that causes ultra-slow spatial
aggregation of subdiffusive particles due to dominance over the standard
advection and diffusion terms, in the long-time limit. This uncovers the
anomalous mechanism by which non-uniform distributions can occur. We perform
experiments on intracellular lysosomal distributions and Monte Carlo
simulations and find excellent agreement between the asymptotic solution,
particle histograms and experiments.Comment: 6 page
Long memory stochastic volatility in option pricing
The aim of this paper is to present a simple stochastic model that accounts
for the effects of a long-memory in volatility on option pricing. The starting
point is the stochastic Black-Scholes equation involving volatility with
long-range dependence. We consider the option price as a sum of classical
Black-Scholes price and random deviation describing the risk from the random
volatility. By using the fact the option price and random volatility change on
different time scales, we find the asymptotic equation for the derivation
involving fractional Brownian motion. The solution to this equation allows us
to find the pricing bands for options
Emergence of L\'{e}vy Walks in Systems of Interacting Individuals
Recent experiments (G. Ariel, et al., Nature Comm. 6, 8396 (2015)) revealed
an intriguing behavior of swarming bacteria: they fundamentally change their
collective motion from simple diffusion into a superdiffusive L\'{e}vy walk
dynamics. We introduce a nonlinear non-Markovian persistent random walk model
that explains the emergence of superdiffusive L\'{e}vy walks. We show that the
alignment interaction between individuals can lead to the superdiffusive growth
of the mean squared displacement and the power law distribution of run length
with infinite variance. The main result is that the superdiffusive behavior
emerges as a nonlinear collective phenomenon, rather than due to the standard
assumption of the power law distribution of run distances from the inception.
At the same time, we find that the repulsion/collision effects lead to the
density dependent exponential tempering of power law distributions. This
qualitatively explains experimentally observed transition from superdiffusion
to the diffusion of mussels as their density increases (M. de Jager et al.,
Proc. R. Soc. B 281, 20132605 (2014))
Nonlinear degradation enhanced transport of morphogens performing subdiffusion
We study a morphogen gradient formation under nonlinear degradation and
subdiffusive transport. In the long time limit we obtain the nonlinear effect
of degradation enhanced diffusion, resulting from the interaction of
non-Markovian subdiffusive transport with a nonlinear reaction. We find the
stationary profile of power-law type, which has implications for robustness,
with the shape of the profile being controlled by the anomalous exponent. Far
away from the source of morphogens, any changes in rate of production are not
felt.Comment: 7 page
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