Norbert Christopeit

Quasi-least-squares estimation in semimartingale regression models

We consider a linear regression model in continuous time with predetermined stochastic regressors and a local martingale as noise process. The method of estimation of the regression parameters is inspired by the classical least-squares estimate in the discrete time setting. It is shown that under a certain condition limiting the growth of the maximal eigenvalue of the design matrix with respect to the minimal eigenvalue, this “quasi-least-squares” estimate converges with probability one to the true parameter values. The proof uses results from semimartingale theory, in particular the transformation formula and some stability properties of local martingales.

WEAK CONVERGENCE OF NONLINEAR TRANSFORMATIONS OF INTEGRATED PROCESSES: THE MULTIVARIATE CASE

We consider weak convergence of sample averages of nonlinearly transformed stochastic triangular arrays satisfying a functional invariance principle. A fundamental paradigm for such processes is constituted by integrated processes. The results obtained are extensions of recent work in the literature to the multivariate and non-Gaussian case. As admissible nonlinear transformation, a new class of functionals (so-called locally p -integrable functions) is introduced that adapts the concept of locally integrable functions in Potscher (2004, Econometric Theory 20, 1–22) to the multidimensional setting.

On the existence and characterization of arbitrage–free measure in contingent claim valuation

In this paper, necessary and sufficient conditions for existence of equivalent martiangles measures in semimartingale models for the pricing of contingent claims are derived

Linear minimax estimation with ellipsoidal constraints

We consider the simultaneous linear minimax estimation problem in linear models with ellipsoidal constraints imposed on an unknown parameter. Using convex analysis, we derive necessary and sufficient optimality conditions for a matrix to define the linear minimax estimator. For certain regions of the set of characteristics of linear models and constraints, we exploit these optimality conditions and get explicit formulae for linear minimax estimators.

On the existence of weak solutions to stochastic differential equations with degenerate diffusion

For a certain class of stochastic differential equations with nonlinear drift and degenerate diffusion term existence of a weak solution is shown.

A Note on an Estimation Problem in Models with Adaptive Learning

This paper provides an example of a linear regression model with predetermined stochastic regressors for which the sufficient condition for strong consistency of the ordinary least squares estimator by Lai & Wei (1982, Annals of Statistics) is not met. Nevertheless, the estimator is strongly consistent, as shown in a companion paper, cf. Christopeit & Massmann (2013b). This is intriguing because the Lai & Wei condition is the best currently available and is referred to as “in some sense the weakest possible”. Moreover, the example discussed in this paper arises naturally in a class of macroeconomic models with adaptive learning, the estimation of which has recently gained popularity amongst researchers and policy makers.

Strong Consistency of the Least-Squares Estimator in Simple Regression Models with Stochastic Regressors

Strong consistency of least squares estimators of the slope parameter in simple linear regression models is established for predetermined stochastic regressors. The main result covers a class of models which falls outside the applicability of what is presently available in the literature. An application to the identification of economic models with adaptive learning is discussed.

Limited risk control of the ornstein-uhleeck process

We consider the problem of steering a system described by a simple one dimensional: linear stochastic differential equation in such a way as to maximize the upper allowable limit that is exceeded by the final state with a prescribed probability. The optimal controls are computed by applying the maximum principle to an equivalent deterministic problem.. This model can be used among other things to find optimal strategies for specific advertising. activities of a firm.