Non parametric estimation of the diffusion coefficients of a diffusion with jumps

In this article, we consider a jump diffusion process ${\left(X_t\right)}_{t\ge 0}$, with drift function $b$, diffusion coefficient $\sigma$ and jump coefficient ${\xi }^2$. This process is observed at discrete times $t=0,\Delta ,\dots ,n\Delta$. The sampling interval $\Delta$ tends to 0 and the time interval $n\Delta$ tends to infinity. We assume that ${\left(X_t\right)}_{t\ge 0}$ is ergodic, strictly stationary and exponentially $\beta$-mixing. We use a penalized least-square approach to compute adaptive estimators of the functions ${\sigma }^2+{\xi }^2$ and ${\sigma }^2$. We provide bounds for the risks of the two estimators.     

Wavelet estimation of the diffusion coefficient in time dependent diffusion models

The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the L~r convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example, in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.     

Non parametric estimation for random walks in random environment

We investigate the problem of estimating the cumulative distribution function (c.d.f.) $F$ of a distribution $\nu$ from the observation of one trajectory of the random walk in i.i.d. random environment with distribution $\nu$ on $\mathbb{Z}$. We first estimate the moments of $\nu$, then combine these moment estimators to obtain a collection of estimators ${\left({\stackrel{?}{F}}_n^M\right)}_{M\ge 1}$ of $F$, our final estimator is chosen among this collection by Goldenshluger–Lepski’s method. This estimator is easily computable. We derive convergence rates for this estimator depending on the Hölder regularity of $F$ and on the divergence rate of the walk. Our rate is minimal when the chain realizes a trade-off between a fast exploration of the sites, allowing to get more information and a larger number of visits of each site, allowing a better recovery of the environment itself.     

Non-parametric estimation of the diffusion coefficient from noisy data

We consider a diffusion process (X _t ) _t????0, with drift b(x) and diffusion coefficient ??(x). At discrete times t _k ?=?k ?? for k from 1 to M, we observe noisy data of the sample path, ${Y_{k\delta}=X_{k\delta}+\varepsilon_{k}}$ . The random variables ${\left(\varepsilon_{k}\right)}$ are i.i.d, centred and independent of (X _t ). The process (X _t ) _t????0 is assumed to be strictly stationary, ??-mixing and ergodic. In order to reduce the noise effect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that ???=?p ?? is small whereas M ?? is large. Then, the diffusion coefficient ?? ² is estimated in a compact set A in a non-parametric way by a penalized least squares approach and the risk of the resulting adaptive estimator is bounded. We provide several examples of diffusions satisfying our assumptions and we carry out various simulations. Our simulation results illustrate the theoretical properties of our estimators.     

Non parametric estimation of smooth stationary covariance functions by interpolation methods

In this paper we introduce a nonparametric approach for the estimation of the covariance function of a stationary stochastic process X _t indexed by The data consist of a finite number of observations of the process at irregularly spaced time points and the aim is to estimate the covariance at any lag point without parametric assumptions and in such a way that it is a positive definite function. After interpolating the process, we use the estimator designed by Parzen (Technometrics 3:167–190,1961) for continuous-time data. Our estimator is shown to be consistent under smoothness assumptions on the covariance. Its performance is evaluated by simulations.     

Parametric estimation for a simple branching diffusion process

A simple branching diffusion process is given as an elementary model of spatial evolution. A parametric estimation theory is presented for this model. As side results, a spatial central limit theorem and spatial strong law of large numbers are also obtained.     

Maximum likelihood estimation for a branching diffusion process

This paper investigates the properties of the maximum likelihood estimators of the drift and diffusion coefficients under three sampling schemes for a branching diffusion process in which the branching process is a linear birth process and the diffusion is in accordance with the Brownian motion with drift.     

Non-parametric adaptive estimation of the drift for a jump diffusion process

In this article, we consider a jump diffusion process _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ observed at discrete times t=0,Δ,…,nΔ$t=0,\Delta ,\dots ,n\Delta$. The sampling interval Δ$\Delta$ tends to 0 and nΔ$n\Delta$ tends to infinity. We assume that _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ is ergodic, strictly stationary and exponentially β$\beta$-mixing. We use a penalised least-square approach to compute two adaptive estimators of the drift function b$b$. We provide bounds for the risks of the two estimators.     

Stationary distribution function estimation for ergodic diffusion process

The problem of nonparametric stationary distribution function estimation by the observation of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found, and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.     

Improved estimation in a non-Gaussian parametric regression

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James–Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.     

On Robust estimation of a correlation coefficient

Robust estimation of the correlation coefficient of a bivariate normal distribution is considered in the case of a contamination scheme. A number of conventional robust estimates are studied, and some new estimates are proposed. Their properties are examined on finite samples and in asymptotics with the use of Monte-Carlo and the influence functions techniques correspondingly. It is shown that one of the proposed estimates called a median correlation coefficient has high robustness properties. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models. Kazan, Russian, 1995, Part II.     

Analytic solution and estimation of parameters on a stochastic exponential model for a technological diffusion process

In this paper we examine the behaviour of a stochastic model that describes a technological diffusion process (continuously increasing process). Furthermore we obtain a solution for the proposed model through the estimation of the volatility using three different approximations. The adjustment of real data to the final stochastic model confirms its ability of describing and forecasting real cases.     

Adaptive estimation of an ergodic diffusion process based on sampled data

We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained.     

Minimizing the variance of a mathematical expectation estimate for a diffusion process functional based on a parametric transformation of a parabolic boundary value problem

This paper deals with finding ways of reducing the variance of a mathematical expectation estimate for the functional of a diffusion process moving in a domain with an absorbing boundary. The estimate of mathematical expectation of the functional is obtained based on a numerical solution of stochastic differential equations (SDEs) by using the Euler method. A formula of the limiting variance is derived with decreasing integration step in the Euler method. A method of reducing the variance value of the estimate based on transformation of the parabolic boundary value problem corresponding to the diffusion process is proposed. Some numerical results are presented.     

Non parametric statistical models for on-line text classification

Social media, such as blogs and on-line forums, contain a huge amount of information that is typically unorganized and fragmented. An important issue, that has been raising importance so far, is to classify on-line texts in order to detect possible anomalies. For example on-line texts representing consumer opinions can be, not only very precious and profitable for companies, but can also represent a serious damage if they are negative or faked. In this contribution we present a novel statistical methodology rooted in the context of classical text classification, in order to address such issues. In the literature, several classifiers have been proposed, among them support vector machine and naive Bayes classifiers. These approaches are not effective when coping with the problem of classifying texts belonging to an unknown author. To this aim, we propose to employ a new method, based on the combination of classification trees with non parametric approaches, such as Kruskal?CWallis and Brunner?CDette?CMunk test. The main application of what we propose is the capability to classify an author as a new one, that is potentially trustable, or as an old one, that is potentially faked.     

Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation

We study parametric optimization with respect to an integral criterion of the higher coefficient and the right-hand side of a second-order semilinear elliptic equation with the Dirichlet boundary condition. We obtain formulas for the first partial derivatives of the objective functional with respect to the control parameters. The total preservation (preservation for the entire set of control parameters) of the unique solvability of the boundary value problem for this equation is proved based on the theory of monotone operators.     

Bipower-type estimation in a noisy diffusion setting

We consider a new class of estimators for volatility functionals in the setting of frequently observed Itō diffusions which are disturbed by i.i.d. noise. These statistics extend the approach of pre-averaging as a general method for the estimation of the integrated volatility in the presence of microstructure noise and are closely related to the original concept of bipower variation in the no-noise case. We show that this approach provides efficient estimators for a large class of integrated powers of volatility and prove the associated (stable) central limit theorems. In a more general Itō semimartingale framework this method can be used to define both estimators for the entire quadratic variation of the underlying process and jump-robust estimators which are consistent for various functionals of volatility. As a by-product we obtain a simple test for the presence of jumps in the underlying semimartingale.     

A test for a parametric form of the volatility in second-order diffusion models

Second-order diffusion models have been found to be promising in analyzing financial market data. Based on nonparametric fitting, Nicolau (Stat Probabil Lett 78(16):2700–2704, 2008) suggested that the quadratic function may be an appropriate specification of the volatility when a second-order diffusion model is used to analyze some European and American financial market data sets, which motivates us to develop a formal statistical test for this finding. To achieve the task, a generalized likelihood ratio test is proposed in this paper and a residual-based bootstrap is suggested to compute the p value of the test. The analysis of many real-world financial market data sets demonstrates that the quadratic specification of the volatility function is in general reasonable.     

A new refinement indicator for adaptive parameterization: Application to the estimation of the diffusion coefficient in an elliptic problem

We propose a new refinement indicator (NRI) for adaptive parameterization to determine the diffusion coefficient in an elliptic equation in two-dimensional space. The diffusion coefficient is assumed to be a piecewise constant space function. The unknowns are both the parameter values and the zonation. Refinement indicators are used to localize parameter discontinuities in order to construct iteratively the zonation (parameterization). The refinement indicator is obtained usually by using the first-order effect on the objective function of removing degrees of freedom for a current set of parameters. In this work, in order to reduce the computation costs, we propose a new refinement indicator based on the second-order effect on the objective function. This new refinement indicator depends on the objective function, and its first and second derivatives with respect to the parameter constraints. Numerical experiments show the high efficiency of the new refinement indicator compared to the standard one.     

Trajectory attractor for a reaction-diffusion system with a small diffusion coefficient

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