5,023 research outputs found
Geometric Stable processes and related fractional differential equations
We are interested in the differential equations satisfied by the density of
the Geometric Stable processes
, with stability \ index and asymmetry
parameter , both in the univariate and in the
multivariate cases. We resort to their representation as compositions of stable
processes with an independent Gamma subordinator. As a preliminary result, we
prove that the latter is governed by a differential equation expressed by means
of the shift operator. As a consequence, we obtain the space-fractional
equation satisfied by the density of For some
particular values of and we get some interesting results
linked to well-known processes, such as the Variance Gamma process and the
first passage time of the Brownian motion.Comment: 12 page
Long-memory Gaussian processes governed by generalized Fokker-Planck equations
It is well-known that the transition function of the Ornstein-Uhlenbeck
process solves the Fokker-Planck equation. This standard setting has been
recently generalized in different directions, for example, by considering the
so-called -stable driven Ornstein-Uhlenbeck, or by time-changing the
original process with an inverse stable subordinator. In both cases, the
corresponding partial differential equations involve fractional derivatives (of
Riesz and Riemann-Liouville types, respectively) and the solution is not
Gaussian. We consider here a new model, which cannot be expressed by a random
time-change of the original process: we start by a Fokker-Planck equation (in
Fourier space) with the time-derivative replaced by a new fractional
differential operator. The resulting process is Gaussian and, in the stationary
case, exhibits a long-range dependence. Moreover, we consider further
extensions, by means of the so-called convolution-type derivative.Comment: 24, accepted for publicatio
Multivariate fractional Poisson processes and compound sums
In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain
some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes
Random-time processes governed by differential equations of fractional distributed order
We analyze here different types of fractional differential equations, under
the assumption that their fractional order is random\ with
probability density We start by considering the fractional extension
of the recursive equation governing the homogeneous Poisson process
\ We prove that, for a particular (discrete) choice of , it
leads to a process with random time, defined as The distribution of the
random time argument can be
expressed, for any fixed , in terms of convolutions of stable-laws. The new
process is itself a renewal and
can be shown to be a Cox process. Moreover we prove that the survival
probability of , as well as its
probability generating function, are solution to the so-called fractional
relaxation equation of distributed order (see \cite{Vib}%).
In view of the previous results it is natural to consider diffusion-type
fractional equations of distributed order. We present here an approach to their
solutions in terms of composition of the Brownian motion with the
random time . We thus provide an
alternative to the constructions presented in Mainardi and Pagnini
\cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order
case.Comment: 26 page
Some Economic Implications of the COVID-19 Pandemic in Nebraska
It is now clear that the spread, mortality and morbidity impacts of the coronavirus pandemic are sizeable but extremely heterogeneous across multiple dimensions. Geography shows that places with lower human con-centrations (urban/rural divide) and away from main travel axes, such as major interstate highways and in-ternational airports, have a lower incidence of cases. Large urban centers with high human concentrations have been much disproportionally affected and with much higher mortality rates. Age and health status are equally important. Mortality increases dramatically for people 60 years old and older. People with comorbidi-ties (cardiovascular, obesity, diabetes, and others) are much more likely to be hospitalized and die of COVID -19 than are healthy people. Family and household composition is also important. Multigenerational households are much more common in say Italy than in Sweden. Swedish households tend to live more inde-pendently often in one-person households, which pro-vides some “cultural” self-isolation, which is helpful in case of a pandemic. In addition, medical infrastructure and preparedness vary greatly across states with devas-tating consequences like in New York City, partly be-cause the pandemic hit early, and partly because of the lack of intensive care unit (ICU) infrastructure (COVID-19 Project). States in the Midwest had more time to prepare and learn to ramp up testing etc
Large deviations for fractional Poisson processes
We prove large deviation principles for two versions of fractional Poisson
processes. Firstly we consider the main version which is a renewal process; we
also present large deviation estimates for the ruin probabilities of an
insurance model with constant premium rate, i.i.d. light tail claim sizes, and
a fractional Poisson claim number process. We conclude with the alternative
version where all the random variables are weighted Poisson distributed.
Keywords: Mittag Leffler function; renewal process; random time ch
The distribution of the local time for "pseudo-processes" and its connections with fractional diffusion equations
We prove that the pseudoprocesses governed by heat-type equations
of order have a local time in zero (denoted by
) whose distribution coincides with the folded
fundamental solution of a fractional diffusion equation of order
: The distribution of is also
expressed in terms of stable laws of order and their
form is analyzed. Furthermore, it is proved that the distribution
of is connected with a wave equation as
. The distribution of the local time in zero
for the pseudoprocess related to the Myiamoto’s equation is also
derived and examined together with the corresponding
telegraph-type fractional equation
Fractional diffusion equations and processes with randomly varying time
In this paper the solutions to fractional diffusion
equations of order are analyzed and interpreted as densities of
the composition of various types of stochastic processes. For the fractional
equations of order , we show that the solutions
correspond to the distribution of the -times iterated Brownian
motion. For these processes the distributions of the maximum and of the sojourn
time are explicitly given. The case of fractional equations of order , is also investigated and related to Brownian motion
and processes with densities expressed in terms of Airy functions. In the
general case we show that coincides with the distribution of Brownian
motion with random time or of different processes with a Brownian time. The
interplay between the solutions and stable distributions is also
explored. Interesting cases involving the bilateral exponential distribution
are obtained in the limit.Comment: Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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