378 research outputs found
Stochastic models associated to a Nonlocal Porous Medium Equation
The nonlocal porous medium equation considered in this paper is a degenerate
nonlinear evolution equation involving a space pseudo-differential operator of
fractional order. This space-fractional equation admits an explicit,
nonnegative, compactly supported weak solution representing a probability
density function. In this paper we analyze the link between isotropic transport
processes, or random flights, and the nonlocal porous medium equation. In
particular, we focus our attention on the interpretation of the weak solution
of the nonlinear diffusion equation by means of random flights.Comment: Published at https://doi.org/10.15559/18-VMSTA112 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
A family of random walks with generalized Dirichlet steps
We analyze a class of continuous time random walks in
with uniformly distributed directions. The steps performed by these processes
are distributed according to a generalized Dirichlet law. Given the number of
changes of orientation, we provide the analytic form of the probability density
function of the position reached, at time
, by the random motion. In particular, we analyze the case of random walks
with two steps.
In general, it is an hard task to obtain the explicit probability
distributions for the process . Nevertheless,
for suitable values for the basic parameters of the generalized Dirichlet
probability distribution, we are able to derive the explicit conditional
density functions of . Furthermore, in some
cases, by exploiting the fractional Poisson process, the unconditional
probability distributions are obtained. This paper extends in a more general
setting, the random walks with Dirichlet displacements introduced in some
previous papers
On a family of test statistics for discretely observed diffusion processes
We consider parametric hypotheses testing for multidimensional ergodic
diffusion processes observed at discrete time. We propose a family of test
statistics, related to the so called -divergence measures. By taking into
account the quasi-likelihood approach developed for studying the stochastic
differential equations, it is proved that the tests in this family are all
asymptotically distribution free. In other words, our test statistics weakly
converge to the chi squared distribution. Furthermore, our test statistic is
compared with the quasi likelihood ratio test. In the case of contiguous
alternatives, it is also possible to study in detail the power function of the
tests.
Although all the tests in this family are asymptotically equivalent, we show
by Monte Carlo analysis that, in the small sample case, the performance of the
test strictly depends on the choice of the function . Furthermore, in
this framework, the simulations show that there are not uniformly most powerful
tests
Random flights connecting Porous Medium and Euler-Poisson-Darboux equations
In this paper we consider the Porous Medium Equation and establish a
relationship between its Kompanets-Zel'dovich-Barenblatt solution u(\xd,t),
\xd\in \mathbb R^d,t>0 and random flights. The time-rescaled version of
u(\xd,t) is the fundamental solution of the Euler-Poisson-Darboux equation
which governs the distribution of random flights performed by a particle whose
displacements have a Dirichlet probability distribution and choosing directions
uniformly on a -dimensional sphere (see, e.g., \cite{dgo}).
We consider the space-fractional version of the Euler-Poisson-Darboux
equation and present the solution of the related Cauchy problem in terms of the
probability distributions of random flights governed by the classical
Euler-Poisson-Darboux equation. Furthermore, this research is also aimed at
studying the relationship between the solutions of a fractional Porous Medium
Equation and the fractional Euler-Poisson-Darboux equation.
A considerable part of the paper is devoted to the analysis of the
probabilistic tools of the solutions of the fractional equations. Also the
extension to higher-order Euler-Poisson-Darboux equation is considered and the
solutions interpreted as compositions of laws of pseudoprocesses
Asymptotic results for random flights
The random flights are (continuous time) random walkswith finite velocity.
Often, these models describe the stochastic motions arising in biology. In this
paper we study the large time asymptotic behavior of random flights. We prove
the large deviation principle for conditional laws given the number of the
changes of direction, and for the non-conditional laws of some standard random
flights.Comment: 3 figure
Divergences Test Statistics for Discretely Observed Diffusion Processes
In this paper we propose the use of -divergences as test statistics to
verify simple hypotheses about a one-dimensional parametric diffusion process
\de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t, from discrete
observations with , , under the asymptotic scheme , and
. The class of -divergences is wide and includes
several special members like Kullback-Leibler, R\'enyi, power and
-divergences. We derive the asymptotic distribution of the test
statistics based on -divergences. The limiting law takes different forms
depending on the regularity of . These convergence differ from the
classical results for independent and identically distributed random variables.
Numerical analysis is used to show the small sample properties of the test
statistics in terms of estimated level and power of the test
On penalized estimation for dynamical systems with small noise
We consider a dynamical system with small noise for which the drift is
parametrized by a finite dimensional parameter. For this model we consider
minimum distance estimation from continuous time observations under
-penalty imposed on the parameters in the spirit of the Lasso approach
with the aim of simultaneous estimation and model selection. We study the
consistency and the asymptotic distribution of these Lasso-type estimators for
different values of . For we also consider the adaptive version of the
Lasso estimator and establish its oracle properties
Change point estimation for the telegraph process observed at discrete times
The telegraph process models a random motion with finite velocity and it is
usually proposed as an alternative to diffusion models. The process describes
the position of a particle moving on the real line, alternatively with constant
velocity or . The changes of direction are governed by an homogeneous
Poisson process with rate In this paper, we consider a change
point estimation problem for the rate of the underlying Poisson process by
means of least squares method. The consistency and the rate of convergence for
the change point estimator are obtained and its asymptotic distribution is
derived. Applications to real data are also presented
Clustering of discretely observed diffusion processes
In this paper a new dissimilarity measure to identify groups of assets
dynamics is proposed. The underlying generating process is assumed to be a
diffusion process solution of stochastic differential equations and observed at
discrete time. The mesh of observations is not required to shrink to zero. As
distance between two observed paths, the quadratic distance of the
corresponding estimated Markov operators is considered. Analysis of both
synthetic data and real financial data from NYSE/NASDAQ stocks, give evidence
that this distance seems capable to catch differences in both the drift and
diffusion coefficients contrary to other commonly used metrics
Empirical -distance test statistics for ergodic diffusions
The aim of this paper is to introduce a new type of test statistic for simple
null hypothesis on one-dimensional ergodic diffusion processes sampled at
discrete times. We deal with a quasi-likelihood approach for stochastic
differential equations (i.e. local gaussian approximation of the transition
functions) and define a test statistic by means of the empirical -distance
between quasi-likelihoods. We prove that the introduced test statistic is
asymptotically distribution free; namely it weakly converges to a
random variable. Furthermore, we study the power under local alternatives of
the parametric test. We show by the Monte Carlo analysis that, in the small
sample case, the introduced test seems to perform better than other tests
proposed in literature
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