Large deviation for lasso diffusion process

Abstract

The aim of the present paper is to extend the large deviation with discontinuous statistics studied in \cite{BDE} to the diffusion $d{\mathbf{x}}^{\epsilon }=-\{{\mathbf{A}}^{\top }(\mathbf{A}{\mathbf{x}}^{\epsilon }-\mathbf{y})+\mu sgn({\mathbf{x}}^{\epsilon })\}dt+\epsilon d\mathbf{w}$. The discontinuity of the drift of the diffusion discussed in \cite{BDE} is equal to the hyperplane $\{\mathbf{x}\in {\mathbb{R}}^d:x_1=0\}$, however, in our case the discontinuity is more complex and is equal to the set $\{\mathbf{x}\in {\mathbb{R}}^d:{\prod }_{i=1}^d{x}_i=0\}$.