Fractional-Boundary-Regularized Deep Galerkin Method for Variational Inequalities in Mixed Optimal Stopping and Control

TL;DR

This paper introduces a novel deep learning method, FBR-DGM, for solving complex variational inequalities in stochastic control, improving accuracy and enabling new benchmarks without analytical solutions.

Contribution

The paper develops a Fractional-Boundary-Regularized Deep Galerkin Method that enhances neural network approximation for variational inequalities in stochastic control problems.

Findings

Enhanced accuracy in neural network solutions.

Enables primal-dual consistency checks as benchmarks.

Improved stability and regularity of solutions.

Abstract

Mixed optimal stopping and stochastic control problems define variational inequalities with non-linear Hamilton-Jacobi-Bellman (HJB) operators, whose numerical solution is notoriously difficult and lack of reliable benchmarks. We first use the dual approach to transform it into a linear operator, and then introduce a Fractional-Boundary-Regularized Deep Galerkin Method (FBR-DGM) that augments the classical $L^2$ loss with Sobolev-Slobodeckij norms on the parabolic boundary, enforcing regularity and yielding consistent improvements in the network approximation and its derivatives. The improved accuracy allows the network to be converted back to the original solution using the dual transform. The self-consistency and stability of the network can be tested by checking the primal-dual relationship among optimal value, optimal wealth, and optimal control, offering innovative benchmarks in the absence of analytical solutions.