313 research outputs found
Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations
It is shown that under a certain condition on a semimartingale and a
time-change, any stochastic integral driven by the time-changed semimartingale
is a time-changed stochastic integral driven by the original semimartingale. As
a direct consequence, a specialized form of the Ito formula is derived. When a
standard Brownian motion is the original semimartingale, classical Ito
stochastic differential equations driven by the Brownian motion with drift
extend to a larger class of stochastic differential equations involving a
time-change with continuous paths. A form of the general solution of linear
equations in this new class is established, followed by consideration of some
examples analogous to the classical equations. Through these examples, each
coefficient of the stochastic differential equations in the new class is given
meaning. The new feature is the coexistence of a usual drift term along with a
term related to the time-change.Comment: 27 pages; typos correcte
Fractional Cauchy problems on bounded domains: survey of recent results
In a fractional Cauchy problem, the usual first order time derivative is
replaced by a fractional derivative. This problem was first considered by
\citet{nigmatullin}, and \citet{zaslavsky} in for modeling some
physical phenomena.
The fractional derivative models time delays in a diffusion process. We will
give a survey of the recent results on the fractional Cauchy problem and its
generalizations on bounded domains D\subset \rd obtained in \citet{m-n-v-aop,
mnv-2}. We also study the solutions of fractional Cauchy problem where the
first time derivative is replaced with an infinite sum of fractional
derivatives. We point out a connection to eigenvalue problems for the
fractional time operators considered. The solutions to the eigenvalue problems
are expressed by Mittag-Leffler functions and its generalized versions. The
stochastic solution of the eigenvalue problems for the fractional derivatives
are given by inverse subordinators
Convolution-type derivatives, hitting-times of subordinators and time-changed -semigroups
In this paper we will take under consideration subordinators and their
inverse processes (hitting-times). We will present in general the governing
equations of such processes by means of convolution-type integro-differential
operators similar to the fractional derivatives. Furthermore we will discuss
the concept of time-changed -semigroup in case the time-change is
performed by means of the hitting-time of a subordinator. We will show that
such time-change give rise to bounded linear operators not preserving the
semigroup property and we will present their governing equations by using again
integro-differential operators. Such operators are non-local and therefore we
will investigate the presence of long-range dependence.Comment: Final version, Potential analysis, 201
Hausdorff dimension of operator semistable L\'evy processes
Let be an operator semistable L\'evy process in \rd
with exponent , where is an invertible linear operator on \rd and
is semi-selfsimilar with respect to . By refining arguments given in
Meerschaert and Xiao \cite{MX} for the special case of an operator stable
(selfsimilar) L\'evy process, for an arbitrary Borel set B\subseteq\rr_+ we
determine the Hausdorff dimension of the partial range in terms of the
real parts of the eigenvalues of and the Hausdorff dimension of .Comment: 23 page
Subdiffusive transport in intergranular lanes on the Sun. The Leighton model revisited
In this paper we consider a random motion of magnetic bright points (MBP)
associated with magnetic fields at the solar photosphere. The MBP transport in
the short time range [0-20 minutes] has a subdiffusive character as the
magnetic flux tends to accumulate at sinks of the flow field. Such a behavior
can be rigorously described in the framework of a continuous time random walk
leading to the fractional Fokker-Planck dynamics. This formalism, applied for
the analysis of the solar subdiffusion of magnetic fields, generalizes the
Leighton's model.Comment: 7 page
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
Inversions of Levy Measures and the Relation Between Long and Short Time Behavior of Levy Processes
The inversion of a Levy measure was first introduced (under a different name)
in Sato 2007. We generalize the definition and give some properties. We then
use inversions to derive a relationship between weak convergence of a Levy
process to an infinite variance stable distribution when time approaches zero
and weak convergence of a different Levy process as time approaches infinity.
This allows us to get self contained conditions for a Levy process to converge
to an infinite variance stable distribution as time approaches zero. We
formulate our results both for general Levy processes and for the important
class of tempered stable Levy processes. For this latter class, we give
detailed results in terms of their Rosinski measures
Semi-Markov Graph Dynamics
In this paper, we outline a model of graph (or network) dynamics based on two
ingredients. The first ingredient is a Markov chain on the space of possible
graphs. The second ingredient is a semi-Markov counting process of renewal
type. The model consists in subordinating the Markov chain to the semi-Markov
counting process. In simple words, this means that the chain transitions occur
at random time instants called epochs. The model is quite rich and its possible
connections with algebraic geometry are briefly discussed. Moreover, for the
sake of simplicity, we focus on the space of undirected graphs with a fixed
number of nodes. However, in an example, we present an interbank market model
where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON
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