arXiv subjects

Alessandro De Gregorio

Publications and source records attributed to Alessandro De Gregorio.

35 records · Page 2Linked to original sources

arXiv2017

On the set of optimal homeomorphisms for the natural pseudo-distance associated with the Lie group S^1

If $φ$ and $ψ$ are two continuous real-valued functions defined on a compact topological space $X$ and $G$ is a subgroup of the group of all homeomorphisms of $X$ onto itself, the natural pseudo-distance $d_G(φ,ψ)$ is defined as the infimum of $\mathcal{L}(g)=\|φ-ψ\circ g \|_\infty$, as $g$ varies in $G$. In this paper, we make a first step towards extending the study of this concept to the case of Lie groups, by assuming $X=G=S^1$. In particular, we study the set of the optimal homeomorphisms for $d_G$, i.e. the elements $ρ_α$ of $S^1$ such that $\mathcal{L}(ρ_α)$ is equal to $d_G(φ,ψ)$. As our main results, we give conditions that a homeomorphism has to meet in order to be optimal, and we prove that the set of the optimal homeomorphisms is finite under suitable conditions.

cs.CG↗

arXiv2014

Reflecting random flights

We consider random flights in $\mathbb{R}^d$ reflecting on the surface of a sphere $\mathbb{S}^{d-1}_R,$ with center at the origin and with radius $R,$ where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere $\mathbb{S}^{d-1}_1$. We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to $n\geq 1$. We show that these distributions involve functions which are solutions of the Euler-Darboux-Poisson equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing $n$ by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.

math.PR↗

arXiv2013

Large deviations for a damped telegraph process

In this paper we consider a slight generalization of the damped telegraph process in Di Crescenzo and Martinucci (2010). We prove a large deviation principle for this process and an asymptotic result for its level crossing probabilities (as the level goes to infinity). Finally we compare our results with the analogous well-known results for the standard telegraph process.

math.PR↗

arXiv2013

A family of random walks with generalized Dirichlet steps

We analyze a class of continuous time random walks in $\mathbb R^d,d\geq 2,$ 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 $\{\underline{\bf X}_d(t),t>0\}$ reached, at time $t>0$, 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 $\{\underline{\bf X}_d(t),t>0\}$ . 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 $\{\underline{\bf X}_d(t),t>0\}$. 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.

math.PR↗

arXiv2013

On the asymptotic behavior of the hyperbolic Brownian motion

The main results in this paper concern large and moderate deviations for the radial component of a $n$-dimensional hyperbolic Brownian motion (for $n\geq 2$) on the Poincaré half-space. We also investigate the asymptotic behavior of the hitting probability $P_η(T_{η_1}^{(n)}<\infty)$ of a ball of radius $η_1$, as the distance $η$ of the starting point of the hyperbolic Brownian motion goes to infinity.

math.PR↗

arXiv2012

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.

math.PR↗

arXiv2012

Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres

Diffusion processes $(\underline{\bf X}_d(t))_{t\geq 0}$ moving inside spheres $S_R^d \subset\mathbb{R}^d$ and reflecting orthogonally on their surfaces $\partial S_R^d$ are considered. The stochastic differential equations governing the reflecting diffusions are presented and their kernels and distributions explicitly derived. Reflection is obtained by means of the inversion with respect to the sphere $S_R^d$. The particular cases of Ornstein-Uhlenbeck process and Brownian motion are examined in detail. The hyperbolic Brownian motion on the Poincarè half-space $\mathbb{H}_d$ is examined in the last part of the paper and its reflecting counterpart within hyperbolic spheres is studied. Finally a section is devoted to reflecting hyperbolic Brownian motion in the Poincarè disc $D$ within spheres concentric with $D$.

math.PR↗

arXiv2011

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 $ϕ$-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 $ϕ$. Furthermore, in this framework, the simulations show that there are not uniformly most powerful tests.

math.ST↗

arXiv2011

On random flights with non-uniformly distributed directions

This paper deals with a new class of random flights $\underline{\bf X}_d(t),t>0,$ defined in the real space $\mathbb{R}^d, d\geq 2,$ characterized by non-uniform probability distributions on the multidimensional sphere. These random motions differ from similar models appeared in literature which take directions according to the uniform law. The family of angular probability distributions introduced in this paper depends on a parameter $ν\geq 0$ which gives the level of drift of the motion. Furthermore, we assume that the number of changes of direction performed by the random flight is fixed. The time lengths between two consecutive changes of orientation have joint probability distribution given by a Dirichlet density function. The analysis of $\underline{\bf X}_d(t),t>0,$ is not an easy task, because it involves the calculation of integrals which are not always solvable. Therefore, we analyze the random flight $\underline{\bf X}_m^d(t),t>0,$ obtained as projection onto the lower spaces $\mathbb{R}^m,m 0.$ Although, in its general framework, the analysis of $\underline{\bf X}_d(t),t>0,$ is very complicated, for some values of $ν$, we can provide some results on the process. Indeed, for $ν=1$, we obtain the characteristic function of the random flight moving in $\mathbb{R}^d$. Furthermore, by inverting the characteristic function, we are able to give the analytic form (up to some constants) of the probability distribution of $\underline{\bf X}_d(t),t>0.$

math.PR↗

arXiv2011

Flying randomly in $\mathbb{R}^d$ with Dirichlet displacements

Random flights in $\mathbb{R}^d,d\geq 2,$ with Dirichlet-distributed displacements and uniformly distributed orientation are analyzed. The explicit characteristic functions of the position $\underline{\bf X}_d(t),\,t>0,$ when the number of changes of direction is fixed are obtained. The probability distributions are derived by inverting the characteristic functions for all dimensions $d$ of $\mathbb{R}^d$ and many properties of the probabilistic structure of $\underline{\bf X}_d(t),t>0,$ are examined. If the number of changes of direction is randomized by means of a fractional Poisson process, we are able to obtain explicit distributions for $P\{\underline{\bf X}_d(t)\in d\underline{\bf x}_d\}$ for all $d\geq 2$. A Section is devoted to random flights in $\mathbb{R}^3$ where the general results are discussed. The existing literature is compared with the results of this paper where in our view the classical Pearson's problem of random flights is resolved by suitably randomizing the step lengths. The random flights where changes of direction are governed by a homogeneous Poisson process are analyzed and compared with the model of Dirichlet-distributed displacements of this work.

math.PR↗

arXiv2009

Stochastic velocity motions and processes with random time

The aim of this paper is to analyze a class of random motions which models the motion of a particle on the real line with random velocity and subject to the action of the friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and the moment generating function of the position reached by the particle at time $t>0$. We are able to derive the explicit probability distributions in few cases for which discuss the connections with the random flights. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we deal with them varying up a random time. Essentially, we consider two different type of random times, namely Bessel and Gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, Exponential). In particular, for the random processes built by means of these compositions, we derive the probability distributions fixed the number of Poisson events. Some remarks on the possible extensions to the random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.

math.PR↗

arXiv2008

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.

q-fin.ST↗

arXiv2008

Divergences Test Statistics for Discretely Observed Diffusion Processes

In this paper we propose the use of $ϕ$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process $\de X_t = b(X_t, θ)\de t + σ(X_t, θ)\de W_t$, from discrete observations $\{X_{t_i}, i=0, ..., n\}$ with $t_i = iΔ_n$, $i=0, 1, >..., n$, under the asymptotic scheme $Δ_n\to0$, $nΔ_n\to\infty$ and $nΔ_n^2\to 0$. The class of $ϕ$-divergences is wide and includes several special members like Kullback-Leibler, Rényi, power and $α$-divergences. We derive the asymptotic distribution of the test statistics based on $ϕ$-divergences. The limiting law takes different forms depending on the regularity of $ϕ$. 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.

math.ST↗

arXiv2007

Renyi information for ergodic diffusion processes

In this paper we derive explicit formulas of the Rényi information, Shannon entropy and Song measure for the invariant density of one dimensional ergodic diffusion processes. In particular, the diffusion models considered include the hyperbolic, the generalized inverse Gaussian, the Pearson, the exponential familiy and a new class of skew-$t$ diffusions.

math.PR↗

arXiv2007

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 $+ v$ or $-v$. The changes of direction are governed by an homogeneous Poisson process with rate $λ>0.$ 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.

math.ST↗

arXiv2007

Parametric estimation for planar random flights observed at discrete times

We deal with a planar random flight $\{(X(t),Y(t)),0<t\leq T\}$ observed at $n+1$ equidistant times $t_i=iΔ_n,i=0,1,...,n$. The aim of this paper is to estimate the unknown value of the parameter $λ$, the underlying rate of the Poisson process. The planar random flights are not markovian, then we use an alternative argument to derive a pseudo-maximum likelihood estimator $\hatλ$ of the parameter $λ$. We consider two different types of asymptotic schemes and show the consistency, the asymptotic normality and efficiency of the estimator proposed. A Monte Carlo analysis for small sample size $n$ permits us to analyze the empirical performance of $\hatλ$. A different approach permits us to introduce an alternative estimator of $λ$ which is consistent, asymptotically normal and asymptotically efficient without the request of other assumptions.

math.ST↗

arXiv2006

Parametric estimation for the standard and geometric telegraph process observed at discrete times

The telegraph process $X(t)$, $t>0$, (Goldstein, 1951) and the geometric telegraph process $S(t) = s_0 \exp\{(μ-\frac12σ^2)t + σX(t)\}$ with $μ$ a known constant and $σ>0$ a parameter are supposed to be observed at $n+1$ equidistant time points $t_i=iΔ_n,i=0,1,..., n$. For both models $λ$, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also $σ>0$ has to be estimated. We propose different estimators of the parameters and we investigate their performance under the high frequency asymptotics, i.e. $Δ_n \to 0$, $nΔ= T<\infty$ as $n \to \infty$, with $T>0$ fixed. The process $X(t)$ in non markovian, non stationary and not ergodic thus we use approximation arguments to derive estimators. Given the complexity of the equations involved only estimators on the first model can be studied analytically. Therefore, we run an extensive Monte Carlo analysis to study the performance of the proposed estimators also for small sample size $n$.

math.ST↗