Entropy-regularized penalization schemes and reflected BSDEs with singular generators

TL;DR

This paper introduces an entropy-regularized penalization scheme for continuous-time optimal stopping problems, providing a smooth approximation of the stopping rule, and analyzes its convergence to solutions of reflected BSDEs with singular generators.

Contribution

It proposes a novel entropy-regularized penalization method for reflected BSDEs, with theoretical analysis and numerical validation in low-dimensional settings.

Findings

The scheme promotes exploration and enables gradient-based learning.

Convergence to solutions of reflected BSDEs with singular generators is established.

Numerical experiments demonstrate the scheme's effectiveness in low-dimensional cases.

Abstract

This paper extends our previous work to continuous-time optimal stopping, focusing on American options in an exploratory setting. Our first contribution is an entropy-regularized penalization scheme, inspired by classical penalization techniques for reflected BSDEs. It yields a smooth approximation of the stopping rule, promotes exploration, and enables gradient-based learning methods. We prove well-posedness, convergence, and illustrate numerical performance in low-dimensional examples. Our second contribution analyzes the behaviour of the scheme as the penalization parameter grows, showing that the limit solves a reflected BSDE with a logarithmically singular generator, for which we establish existence and uniqueness via a monotone limit argument.