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In this article, we study the explosion time of the solution to autonomous stochastic differential equations driven by the fractional Brownian motion with Hurst parameter $H>1/2$. With the help of the Lamperti transformation, we are able to tackle the case of non-constant diffusion coefficients not covered in the literature. In addition, we provide an adaptive Euler-type numerical scheme for approximating the explosion time., Comment: 21 pages

Geographical and Temporal Weighted Regression (GTWR) model is an important local technique for exploring spatial heterogeneity in data relationships, as well as temporal dependence due to its high fitting capacity when it comes to real data. In this article, we consider a GTWR model driven by a spatio-temporal noise, colored in space and fractional in time. Concerning the covariates, we consider that they are correlated, taking into account two interaction types between covariates, weak and strong interaction. Under these assumptions, Weighted Least Squares Estimator (WLS) is obtained, as well as its rate of convergence. In order to evidence the good performance of the estimator studied, it is provided a simulation study of four different scenarios, where it is observed that the residuals oscillate with small variation around zero. The STARMA package of the R software allows obtaining a variant of the $R^{2}$ coefficient, with values very close to 1, which means that most of the variability is explained by the model., Comment: 24 pages, 35 figures

Introduction: The goal of this paper is to study inequities in breast cancer (BC) health care outcomes for Chilean women, including case fatality (FR) and survival rates (SR), stratified by type of health care provider and geographical area. A secondary goal is to estimate BC incidence (IR) and mortality (MR) rates by health care providers and region. Methods: We used two public anonymized databases provided by the Ministry of Health: the national death and hospital discharges datasets. For survival analysis, we used the Kaplan Meier product-limit estimator (KM) with a 95% ci and the Cox proportional hazards model (CM) with null-hypothesis significance testing of p>0.001. Results: We considered a cohort of 58,254 and 16,615 BC hospital discharges and deaths for the period 2007-2018. New cases and deaths due to BC increased by 43.6% and 33.6% respectively. Avg age-adjusted IR and MR were 44.0 and 10.5, respectively. Women affiliated to a private provider (ISAPRE) have an avg age adjusted IR of 60.6 compared to 38.8 for women affiliated with the public provider (FONASA). The national FR has remained constant over time, with a mean of 26.8. Women affiliated with ISAPRE had a considerably lower FR during the period under study, with an avg of 15.7 compared to 27.5 for women in FONASA. The avg 5-year SR were 0.81 and 0.90 for FONASA and ISAPRE. Women from the Metropolitan area have higher SRs than women from other regions. SRs obtained using the CM have a similar behavior to those obtained by the KM. Discussion: Despite the inclusion of BC in the GES plan in 2005 to provide greater, there are still significant differences in FR and SR for patients affiliated to ISAPRE compared to those in FONASA, a choice that is directly associated with socioeconomic level, and for patients in the Metropolitan and other regions. Further studies are required to determine the causes of these disparities., Comment: 18 pages including bibliography, 13 figures

In this article we study the asymptotic behaviour of the least square estimator in a linear regression model based on random observation instances. We provide mild assumptions on the moments and dependence structure on the randomly spaced observations and the residuals under which the estimator is strongly consistent. In particular, we consider observation instances that are negatively superadditive dependent within each other, while for the residuals we merely assume that they are generated by some continuous function. In addition, we prove that the rate of convergence is proportional to the sampling rate $N$, and we complement our findings with a simulation study providing insights on finite sample properties., Comment: Final version will be published in Statistics

In this article, we study the limit distribution of the least square estimator, properly normalized, from a regression model in which observations are assumed to be finite ($\alpha N$) and sampled under two different random times. Based on the limit behavior of the characteristic function and convergence result we prove the asymptotic normality for the least square estimator. We present simulation results to illustrate our theoretical results.

Generalisations of the Ornstein-Uhlenbeck process defined through Langevin equation $dU_t = - \Theta U_t dt + dG_t,$ such as fractional Ornstein-Uhlenbeck processes, have recently received a lot of attention in the literature. In particular, estimation of the unknown parameter $\Theta$ is widely studied under Gaussian stationary increment noise $G$. Langevin equation is well-known for its connections to physics. In addition to that, motivation for studying Langevin equation with a general noise $G$ stems from the fact that the equation characterises all univariate stationary processes. Most of the literature on the topic focuses on the one-dimensional case with Gaussian noise $G$. In this article, we consider estimation of the unknown model parameter in the multidimensional version of the Langevin equation, where the parameter $\Theta$ is a matrix and $G$ is a general, not necessarily Gaussian, vector-valued process with stationary increments. Based on algebraic Riccati equations, we construct an estimator for the matrix $\Theta$. Moreover, we prove the consistency of the estimator and derive its limiting distribution under natural assumptions. In addition, to motivate our work, we prove that the Langevin equation characterises all stationary processes in a multidimensional setting as well., Comment: Alignment of equations has been changed so that they fit inside the margins. Updated reference list

We study one-dimensional stochastic differential equations of form $dX_t = \sigma(X_t)dY_t$, where $Y$ is a suitable H\"older continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac12$. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients $\sigma$ for which we assume very mild conditions. In particular, we allow $\sigma$ to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.

In this article we introduce and study oscillating Gaussian processes defined by $X_t = \alpha_+ Y_t {\bf 1}_{Y_t >0} + \alpha_- Y_t{\bf 1}_{Y_t<0}$, where $\alpha_+,\alpha_->0$ are free parameters and $Y$ is either stationary or self-similar Gaussian process. We study the basic properties of $X$ and we consider estimation of the model parameters. In particular, we show that the moment estimators converge in $L^p$ and are, when suitably normalised, asymptotically normal.

In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{n\in\mathbb{N}}$ of solutions to RDEs with unbounded drifts $(\psi_n)_{n\in\mathbb{N}}$. The penalisation $\psi_n$ increases with $n$. Along the way, we thus also provide an existence theorem and a Doss-Sussmann representation for RDEs with a drift growing at most linearly. In addition, a speed of convergence of the sequence of penalised paths to the reflected solution is obtained. \\ We finally use the penalisation method to prove that the law at time $t>0$ of some reflected Gaussian RDE is absolutely contiuous with respect to the Lebesgue measure.

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 performance of the estimator under different values of the Hurst parameter H., Comment: 19 pages, 4 figures

We study a generalized ARCH model with liquidity given by a general stationary process. We provide minimal assumptions that ensure the existence and uniqueness of the stationary solution. In addition, we provide consistent estimators for the model parameters by using AR(1) type characterisation. We illustrate our results with several examples and simulation studies.

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 illustrate our results., Comment: 18 pages, 3 figures, 4 tables

In this paper we study a stochastic differential equation driven by a fractional Brownian motion with a discontinuous coefficient. We also give an approximation to the solution of the equation. This is a first step to define a fractional version of the skew Brownian motion.

We give a strong approximation of Rosenblatt process via transport processes and we give the rate of convergence.

We study the asymptotic behavior of the maximum likelihood estimator corresponding to the observation of a trajectory of a Skew Brownian motion, through a uniform time discretization. We characterize the speed of convergence and the limiting distribution when the step size goes to zero, which in this case are non-classical, under the null hypothesis of the Skew Brownian motion being an usual Brownian motion. This allows to design a test on the skewness parameter. We show that numerical simulations that can be easily performed to estimate the skewness parameter, and provide an application in Biology., Comment: 26 pages, 8 figures

We consider statistical models driven by Gaussian and non-Gaussian self-similar processes with long memory and we construct maximum likelihood estimators (MLE) for the drift parameter. Our approach is based on the approximation by random walks of the driving noise. We study the asymptotic behavior of the estimators and we give some numerical simulations to illustrate our results., Comment: To appear in "Statistics"

A new Bayesian significance test is adjusted for jump detection in a diffusion process. This is an advantageous procedure for temporal data having extreme valued outliers, like financial data, pluvial or tectonic forces records and others., Comment: 15 pages, 7 figures; real data analysis added

In this paper, we prove a Donsker type approximation theorem for the Rosenblatt process, which is a selfsimilar stochastic process exhibiting long range dependence. By using numerical results and simulated data, we show that this approximation performs very well. We use this result to construct a binary market model driven by this process and we show that the model admits arbitrage opportunities.