Hector Andres Araya

Adolfo Ibáñez University · Facultad de Ingeniería y Ciencias

We consider the problem of the drift parameter estimation for a non-Gaussian long memory Ornstein–Uhlenbeck process driven by a Hermite process. To estimate the unknown parameter, discrete time high-frequency observations at regularly spaced time points and the least squares estimation method are used. By means of techniques based on Wiener chaos a...

In this article, we introduce a non Gaussian long memory process constructed by the aggregation of independent copies of a fractional Lévy Ornstein-Uhlenbeck process with random coefficients. Several properties and a limit theorem are studied for this new process. Finally, some simulations of the limit process are shown.

In this article, we consider the problem of parameter estimation in a power-type diffusion driven by fractional Brownian motion with Hurst parameter in (1/2, 1). To estimate the parameters of the process, we use an approximate bayesian computation method. Also, a particular case is addressed by means of variations and wavelet-type methods.Several t...

We consider the problem of drift parameter estimation in a stochastic differential equation driven by fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} and small diffusion. The technique that we used is the trajectory fitting method. Strong consistency and asymptotic distribution of the estimator are established...

n numerous applications, data are observed at random times. Our main purpose is to study a model observed at random times that incorporates a long-memory noise process with a fractional Brownian Hurst exponent H. We propose a least squares estimator in a linear regression model with long-memory noise and a random sampling time called "jittered samp...

This article studies the monthly variability of anchovy (Engraulis ringens) in northern Chile, related with the environmental effect of sea surface temperature on the landings of the fishery. In order to achieve that goal, a variant of the autoregressive conditional heteroskedastic (ARCH) model is proposed, in which an additional covariate is inclu...

In this study, we prove the strong consistency of the least squares estimator in a random sampled linear regression model with long-memory noise and an independent set of random times given by renewal process sampling. Additionally, we illustrate how to work with a random number of observations up to time T = 1. A simulation study is provided to il...

In this work, we introduce a new process by modifying the kernel in the time domain representation of the generalized Hermite process. This modification is constructed by means of multiplication of the kernel in the time definition of the process by an exponential tempering factor {\lambda} > 0 such that this new process is well defined. Several pr...

In this article, we propose an Euler type scheme to approximate the solutions of stochastic differential equations with non-lipschitz diffusion coefficient driven by fractional Brownian motion (fBm) with Hurst parameter 1/2 < H < 1. Using fractional calculus tools and pathwise integration respect to fBm, the obtained rate of convergence is n^α(1−2H...

This paper deals with the problem of parameter estimation in a class of stochastic differential equations driven by a fractional Brownian motion with H≥1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsi...

In this article, we study the parametric problem of estimating the coefficient for a discrete time model driven by a fractional Poisson noise, when high-frequency observations are given. We consider weighted least squares and maximum likelihood estimators. Thus, asymptotic behavior of the estimators is proved and a simulation study is shown to illu...

In this paper, we study Hermite spatial variations for the solution to the stochastic heat equation with space-time white noise. We prove that these variations satisfy the central limit theorem and we obtain the almost sure central limit theorem

In this article, we introduce a non Gaussian long memory process constructed by the aggregation of independent copies of a fractional L\'evy Ornstein-Uhlenbeck process with random coefficients. Several properties and a limit theorem are studied for this new process. Finally, some simulations of the limit process are shown.

Whale-watching (WW) is an activity which has been increasing worldwide due to the great interest of tourists and the economic benefits it provides to local communities. However, it has been reported that this activity affects the behavioral patterns of some cetaceans, although for some species such as the fin whale (Balaenoptera physalus) this has...

Whale-watching (WW) is an activity which has been increasing worldwide due to the great interest of tourists and the economic benefits it provides to local communities. However, it has been reported that this activity affects the behavioral patterns of some cetaceans, although for some species such as the fin whale (Balaenoptera physalus) this has...

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1/2<H<1. Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fraction...

We consider the sequence of spatial quadratic variations of the solution to the stochastic heat equation with space-time white noise. This sequence satisfies a Central Limit Theorem. By using Malliavin calculus, we refine this result by proving the convergence of the sequence of densities and by finding the second-order term in the asymptotic expan...

In this short note, we give the representation of the non symmetric Rosenblatt process as a Wiener–Itô multiple integral with respect to the Brownian motion on a finite interval. Based on this representation, we obtain a least square-type estimator for an unknown parameter of the drift coefficient of a simple model driven by the non symmetric Rosen...

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (1/4, 1/2). Towards this end, we apply Doss-Sussmann representation of the solution and an approximation of this representation using a first order Taylor expansion. The obtained rate of convergence is n^(...

In this work we present a group of theoretical models for reaction times arising from simple-choice task tests. In particular, we argue for the inclusion of a shifted version of the Gamma distribution as a theoretical model based on a mathematical result on first hitting times. We contrast the goodness-of-fit of those models with the Ex-Gaussian di...

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter H in (1/4; 1/2). Towards this end, we apply Doss-Sussmann representation of the solution and an approximation of this representation using a first order Taylor expansion. The obtained rate of convergence is n^...

In this article, we present the least squares estimator for the drift parameter in a linear regression model driven by the increment of a fractional Brownian motion sampled at random times. For two different random times, Jittered and renewal process sampling, consistency of the estimator is proven. A simulation study is provided to illustrate the...

In this article, we present the least squares estimator for the drift parameter in a linear regression model driven by the increment of a fractional Brownian motion sampled at random times. For two different random times, Jittered and renewal process sampling, consistency of the estimator is proven. A simulation study is provided to illustrate the...

In this paper we study a Donsker type theorem for the fractional Poisson process (fPp). We present the random walk discretization and its associated convergence theorem in the Skorohod topology. Simulation results are also presented.

We consider a d-parameter Hermite process with Hurst index [Formula presented] and we study its limit behavior in distribution when the Hurst parameters Hi,i=1,.,d (or a part of them) converge to [Formula presented] and/or 1. The limit obtained is Gaussian (when at least one parameter tends to [Formula presented] and non-Gaussian (when at least one...

In this article, we study the problem of parameter estimation for a discrete Ornstein - Uhlenbeck model driven by Poisson fractional noise. Based on random walk approximation for the noise, we study least squares and maximum likelihood estimators. Thus, asymptotic behaviours of the estimator is carried out, and a simulation study is shown to illust...