Large deviation for lasso diffusion process

TL;DR

This paper extends large deviation principles to a diffusion process with a complex discontinuous drift, generalizing previous work to a more intricate set of discontinuities involving multiple coordinate hyperplanes.

Contribution

It introduces a large deviation analysis for a diffusion with a discontinuous drift set defined by the product of coordinates, expanding the scope of prior results.

Findings

Extended large deviation results to complex discontinuous drifts

Analyzed diffusion with discontinuity set as coordinate hyperplanes

Generalized previous models to higher-dimensional cases

Abstract

The aim of the present paper is to extend the large deviation with discontinuous statistics studied in \cite{BDE} to the diffusion $d\mathbf{x}^\varepsilon = -\{\mathbf{A}^\top (\mathbf{A} \mathbf{x}^\varepsilon - \mathbf{y}) + \mu sgn(\mathbf{x}^\varepsilon)\}dt + \varepsilon d\mathbf{w}$. 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 $\{\mathbf{x} \in \mathbb{R}^d:\ \prod_{i=1}^dx_i=0\}$.