On Lasso estimator for the drift function in diffusion models

TL;DR

This paper analyzes the properties of the Lasso estimator for the drift function in high-dimensional diffusion models, providing theoretical bounds and inequalities under sparsity assumptions.

Contribution

It introduces an oracle inequality and error bounds for the Lasso estimator in multivariate diffusion models with large dimensions.

Findings

Established an oracle inequality for the Lasso estimator.

Derived an $L^2$-error bound using concentration inequalities.

Applied empirical process theory and chaining methods.

Abstract

In this paper we study the properties of the Lasso estimator of the drift component in the diffusion setting. More specifically, we consider a multivariate parametric diffusion model $X$ observed continuously over the interval $[0,T]$ and investigate drift estimation under sparsity constraints. We allow the dimensions of the model and the parameter space to be large. We obtain an oracle inequality for the Lasso estimator and derive an error bound for the $L^2$-distance using concentration inequalities for linear functionals of diffusion processes. The probabilistic part is based upon elements of empirical processes theory and, in particular, on the chaining method.