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

Yun Zhao, Harry Zheng

TL;DR

The paper tackles variational inequalities from mixed optimal stopping and stochastic control by employing a dual transformation to obtain a linear VI and proposing a Fractional-Boundary-Regularized Deep Galerkin Method (FBR-DGM). FBR-DGM augments the standard $L^2$ loss with Sobolev-Slobodeckij norms on the parabolic boundary to enforce boundary regularity and enhance interior derivative accuracy, enabling reliable reconstruction of the primal value via $V(t,x)=\inf_{y>0}\{\widetilde{V}(t,y)+xy\}$. Through extensive numerical experiments, the authors show that FBR-DGM outperforms traditional dual-based methods (BTM, GCA) and the baseline DGM, with strong self-consistency between the primal value, optimal wealth, and optimal control, especially for power utility. The work offers a practical, efficient framework for solving high-dimensional mixed stopping/control problems and provides a foundation for further extensions to more complex financial settings and theoretical convergence analysis.

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.

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

TL;DR

The paper tackles variational inequalities from mixed optimal stopping and stochastic control by employing a dual transformation to obtain a linear VI and proposing a Fractional-Boundary-Regularized Deep Galerkin Method (FBR-DGM). FBR-DGM augments the standard loss with Sobolev-Slobodeckij norms on the parabolic boundary to enforce boundary regularity and enhance interior derivative accuracy, enabling reliable reconstruction of the primal value via . Through extensive numerical experiments, the authors show that FBR-DGM outperforms traditional dual-based methods (BTM, GCA) and the baseline DGM, with strong self-consistency between the primal value, optimal wealth, and optimal control, especially for power utility. The work offers a practical, efficient framework for solving high-dimensional mixed stopping/control problems and provides a foundation for further extensions to more complex financial settings and theoretical convergence analysis.

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 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.

Paper Structure

This paper contains 15 sections, 1 theorem, 41 equations, 3 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries - mixed optimal stopping and control problem
  4. Main methods
  5. Traditional algorithms
  6. Deep neural network methods (DGM and FBR-DGM)
  7. Numerical experiments
  8. Comparison among approximations of optimal value
  9. Consistency between optimal value and optimal wealth
  10. Consistency between optimal value and optimal control
  11. Conclusion
  12. Proof of Theorem \ref{['thm: verification theorem']}
  13. Algorithms
  14. Algorithm for Section \ref{['sec:numerical 1']} and implementation details
  15. Full Algorithms for Section \ref{['sec:numerical 2']} and Section \ref{['sec:numerical 3']}

Key Result

Theorem 1

Let $\widetilde{V}$ be the solution to eq: VI of tilde V such that $\widetilde{V} \in C^1(Q_y) \cap C(\overline{Q}_y)$, $\widetilde{V} \in C^{1,2,2}(Q_y \backslash \partial C_y)$ with locally bounded derivatives near $\partial C_y$, $|\widetilde{V}(t,y)| \leq C(y^q+1)$ for some constant $C>0$ and $q Moreover, for $s \geq t$, the optimal stopping time is $\tau^* = \inf\{s \geq t \ | (s,Y_s) \in S_y

Figures (3)

  • Figure 1: Approximation quality of DGM, FBR-DGM, and GCA compared with BTM. Panels (a)-(b) show the absolute relative errors for two utilities, averaged over five training seeds.
  • Figure 2: Results of Example \ref{['example: 2']} for networks trained with DGM and FBR-DGM, averaged over seeds.
  • Figure 3: Results of Example \ref{['example: 3']} for networks trained with DGM and FBR-DGM, averaged over seeds.

Theorems & Definitions (6)

  • Theorem 1: Verification Theorem
  • proof
  • Example 1
  • Example 2
  • Example 3
  • proof