Optimal Estimation of Brownian Penalized Regression Coefficients

TL;DR

This paper develops a novel methodology for optimally estimating coefficients in penalized functional regression models where variables evolve over time and are governed by stochastic differential equations, using path integral and Schrödinger-type equations.

Contribution

Introduces a new approach combining stochastic differential equations, penalized regression, and path integral methods to derive closed-form solutions for functional regression coefficients.

Findings

Derived a Schrödinger-type equation encapsulating the system's information.

Obtained closed-form solutions for regression coefficients under various error dynamics.

Provided a framework for optimal penalized functional regression with stochastic processes.

Abstract

In this paper we introduce a new methodology to determine an optimal coefficient of penalized functional regression. We assume the dependent, independent variables and the regression coefficients are functions of time and error dynamics follow a stochastic differential equation. First we construct our objective function as a time dependent residual sum of square and then minimize it with respect to regression coefficients subject to different error dynamics such as LASSO, group LASSO, fused LASSO and cubic smoothing spline. Then we use Feynman-type path integral approach to determine a Schr\"odinger-type equation which have the entire information of the system. Using first order conditions with respect to these coefficients give us a closed form solution of them.