We consider the problem of parameter estimation for multidimensional continuous-time linear stochastic regression models with an arbitrary finite number of unknown parameters and with martingale noise. The main result of the paper claims that the unknown parameters can be estimated with prescribed mean-square precision in this general model providing a unified description of both discrete and continuous time process. Among the conditions on the regressors there is one bounding the growth of the maximal eigenvalue of the design matrix with respect to its minimal eigenvalue. This condition is slightly stronger as compared with the corresponding conditions usually imposed on the regressors in asymptotic investigations but still it enables one to consider models with different behavior of the eigenvalues. The construction makes use of a two-step procedure based on the modified least-squares estimators and special stopping rules.

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On sequential estimation of parameters in semimartingale regression models with continuous time parameter

We study the asymptotic behaviour of the process d M ¢ m 1 M , where M is an R n -continuous vector local martingale and d M ¢ m 1 is the inverse of its predictable quadratic variation (a matrix-valued process). We also give an application (strong consistency and rate of convergence) to multiple linear regression.

About the asymptotic behaviour of continuous vector-valued local martingales and application in multiple linear regression models

This article deals with the problem of parameter estimation of a continuous-time p-dimensional Gaussian autoregressive process. In the stable case, we combine averaging and weighting methods to establish, for the weighted LS-estimator (least-squares estimator) of θ, an almost-sure central limit theorem (ASCLT), a quadratic strong law (QSL) and a central limit theorem (CLT) associated to the QSL with arithmetic convergence rates. In the unstable case, we establish for the LS-estimator of θ, the same optimal asymptotic properties with arithmetic convergence rates. In both cases, using the QSL property, we construct a new estimator of the noise covariance σ2 and we specify its weak convergence rates.

Asymptotic Properties of the LS-estimator of a Gaussian Autoregressive Process by an Averaging Method

The problem of parameter estimation is considered for - continuous-time regression models. A sequential plan is constructed which enables the estimation of unknown parameters with a given precision. The construction uses a two-steps procedure based on modified least-square estimators and particular stopping rules. An important example of the kind of models to which the method applies is the continuous-time unstable autoregression process (for which all roots of the characteristic polynomial have either negative or zero real parts).

On Sequential Estimation of Parameters for Linear Regression with Martingale Noise

Linear multidimensional stochastic approximation procedures in continuous time with martingale errors are considered. An asymptotic behavior of the estimator obtained by trajectory averaging is studied. An asymptotics of integrated squared deviations functionals of the averaged estimator is found. Some results concerning fixed-size confidence regions are presented.

A.R. Darwich

Papers

About the asymptotic behaviour of multidimensional Gaussian martingales and estimates in normal linear regression