The Dantzig selector for a linear model of diffusion processes

TL;DR

This paper applies the Dantzig selector to estimate drift matrices in high-dimensional diffusion process models, establishing consistency and asymptotic normality of the estimators under sparse conditions.

Contribution

It introduces the use of the Dantzig selector for drift estimation in diffusion processes and proves its consistency and asymptotic normality in high-dimensional, sparse settings.

Findings

Proves $l_q$ norm consistency of the estimator.

Establishes variable selection consistency.

Constructs an asymptotically normal estimator.

Abstract

In this paper, a linear model of diffusion processes with unknown drift and diagonal diffusion matrices is discussed. We will consider the estimation problems for unknown parameters based on the discrete time observation in high-dimensional and sparse settings. To estimate drift matrices, the Dantzig selector which was proposed by Cand\'es and Tao in 2007 will be applied. Then, we will prove two types of consistency of the estimator of drift matrix; one is the consistency in the sense of $l_q$ norm for every $q \in [1,\infty]$ and the other is the variable selection consistency. Moreover, we will construct an asymptotically normal estimator of the drift matrix by using the variable selection consistency of the Dantzig selector.