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arxiv: 2405.11392 · v2 · submitted 2024-05-18 · 💱 q-fin.MF · q-fin.CP

Deep Penalty Methods: A Class of Deep Learning Algorithms for Solving High Dimensional Optimal Stopping Problems

Yunfei Peng , Pengyu Wei , Wei Wei This is my paper

Pith reviewed 2026-05-24 00:56 UTC · model grok-4.3

classification 💱 q-fin.MF q-fin.CP
keywords deep learningoptimal stoppingpenalty methodAmerican optionshigh-dimensional PDEDeep BSDEfree boundary problemsoption pricing
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The pith

A penalty approximation to the free-boundary PDE combined with Deep BSDE solves high-dimensional optimal stopping problems with explicit error bounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Deep Penalty Method for high-dimensional optimal stopping problems. It converts the free-boundary PDE into a penalized equation and then applies the Deep BSDE framework to solve that equation. The resulting error is shown to be bounded by the training loss plus terms of order 1 over lambda, lambda times h, and square root of h. Readers care because classical grid methods become unusable once the number of dimensions grows, while this approach remains computationally feasible for tasks such as pricing American options on many assets.

Core claim

The Deep Penalty Method approximates the penalized version of the free-boundary PDE that arises from an optimal stopping problem by means of the Deep BSDE method. The total approximation error is bounded by the value of the loss function together with O(1/λ) + O(λ h) + O(√h), where λ is the penalty parameter and h is the time-step size. Numerical experiments on high-dimensional American option pricing confirm that the algorithm produces accurate prices and exercise strategies at reasonable computational cost.

What carries the argument

The penalized PDE obtained by adding a penalty term that enforces the obstacle condition, solved via the Deep BSDE framework.

If this is right

  • High-dimensional American options become solvable by a single deep-learning procedure rather than by grids or trees.
  • The discretization component of the error converges at rate one-half in the time step.
  • The penalty parameter λ must be chosen in relation to the time step h to keep the overall error small.
  • The same procedure applies to any optimal stopping problem whose penalized PDE can be cast in semilinear form.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same penalty-plus-Deep-BSDE construction may apply to other free-boundary problems that can be written as variational inequalities.
  • Replacing the underlying Deep BSDE solver with a higher-order scheme could improve the observed convergence rate beyond one-half.
  • The method supplies a template for turning other variational inequalities into trainable deep-learning problems.

Load-bearing premise

The Deep BSDE solver approximates the solution of the penalized PDE with an error that adds to the penalty and discretization contributions in the stated way.

What would settle it

Running the algorithm on a low-dimensional optimal stopping problem whose exact solution is known and checking whether the observed error stays within the predicted bound for chosen values of λ and h.

Figures

Figures reproduced from arXiv: 2405.11392 by Pengyu Wei, Wei Wei, Yunfei Peng.

Figure 1
Figure 1. Figure 1: Data flow for the neural network for Z. 13 [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Training loss convergence across dimensions [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Robust Convergence Analysis of the DPM Solver across dimensions [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical Performance Comparison between MSE and [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
read the original abstract

We propose a deep learning algorithm for high dimensional optimal stopping problems. Our method is inspired by the penalty method for solving free boundary PDEs. Within our approach, the penalized PDE is approximated using the Deep BSDE framework proposed by \cite{weinan2017deep}, which leads us to coin the term "Deep Penalty Method (DPM)" to refer to our algorithm. We show that the error of the DPM can be bounded by the loss function and $O(\frac{1}{\lambda})+O(\lambda h) +O(\sqrt{h})$, where $h$ is the step size in time and $\lambda$ is the penalty parameter. This finding emphasizes the need for careful consideration when selecting the penalization parameter and suggests that the discretization error converges at a rate of order $\frac{1}{2}$. We validate the efficacy of the DPM through numerical tests conducted on a high-dimensional optimal stopping model in the area of American option pricing. The numerical tests confirm both the accuracy and the computational efficiency of our proposed algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes the Deep Penalty Method (DPM) for high-dimensional optimal stopping problems. It combines the classical penalty method for free-boundary PDEs with the Deep BSDE solver of Weinan et al. (2017), states an a-priori error bound of the form training loss + O(1/λ) + O(λ h) + O(√h), and reports numerical confirmation on a high-dimensional American put pricing example.

Significance. If the error bound can be established with explicit control on the λ-dependence of the Deep BSDE approximation error, the result would supply a theoretically grounded deep-learning route to high-dimensional optimal stopping, a setting where grid-based and Monte-Carlo methods suffer from the curse of dimensionality.

major comments (2)
  1. [Abstract] Abstract: the claimed error bound is stated without derivation, without explicit assumptions on the Deep BSDE approximation error, and without any quantitative error tables or baseline comparisons. Because the bound is the central theoretical claim, its absence leaves the main result unsupported.
  2. [Abstract] Abstract: the stated combination error ≤ loss + O(1/λ) + O(λ h) + O(√h) presupposes that the Deep BSDE approximation error for the penalized PDE remains bounded independently of λ. Standard Deep BSDE analysis yields constants that grow with the Lipschitz constant of the driver; the penalty term λ max(u-g,0) makes this Lipschitz constant scale at least linearly with λ, so extra factors such as λ√h cannot be ruled out a priori.
minor comments (1)
  1. The numerical section should report concrete values of the training loss, the chosen λ, the observed L² or sup-norm errors, and at least one standard baseline (e.g., Longstaff–Schwartz or a plain Deep BSDE without penalty) so that the practical gain can be quantified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claimed error bound is stated without derivation, without explicit assumptions on the Deep BSDE approximation error, and without any quantitative error tables or baseline comparisons. Because the bound is the central theoretical claim, its absence leaves the main result unsupported.

    Authors: The abstract is a concise summary. The full derivation of the error bound together with the explicit assumptions on the Deep BSDE approximation error appears in Section 3. Section 4 contains numerical error tables for the high-dimensional American put. We will revise the abstract to reference the assumptions and the location of the proof, and we will add further baseline comparisons in the numerical experiments. revision: yes

  2. Referee: [Abstract] Abstract: the stated combination error ≤ loss + O(1/λ) + O(λ h) + O(√h) presupposes that the Deep BSDE approximation error for the penalized PDE remains bounded independently of λ. Standard Deep BSDE analysis yields constants that grow with the Lipschitz constant of the driver; the penalty term λ max(u-g,0) makes this Lipschitz constant scale at least linearly with λ, so extra factors such as λ√h cannot be ruled out a priori.

    Authors: We agree that the λ-dependence of the Deep BSDE error must be controlled explicitly. Our analysis tracks the Lipschitz constant of the penalized driver and absorbs its effect into the O(λ h) term; the resulting bound on the Deep BSDE error does not introduce additional factors of λ beyond those already stated. We will add a clarifying remark (or short appendix) that makes this control explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; error bound is standard decomposition

full rationale

The paper derives an error bound for its DPM by combining the known penalty-method approximation error O(1/λ) + O(λ h) with the training loss of the externally cited Deep BSDE solver (Weinan et al. 2017) plus O(√h) discretization. This is a conventional a-priori bound that does not equate the final claim to its inputs by construction, nor does it rely on self-citations, fitted parameters renamed as predictions, or smuggled ansatzes. The derivation remains self-contained against the cited external framework and does not reduce the stated result to tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the Deep BSDE approximation for the penalized PDE and on the standard convergence properties of the penalty method as λ tends to infinity; no new entities are introduced.

free parameters (2)
  • penalty parameter λ
    User-chosen parameter that appears in both the error bound and the penalized PDE; its value must be selected to balance the O(1/λ) and O(λ h) terms.
  • time step h
    Discretization parameter whose square-root scaling appears in the stated error bound.
axioms (2)
  • domain assumption The Deep BSDE method of Weinan et al. (2017) can be applied directly to the penalized PDE and produces an approximation whose error combines additively with the penalty and discretization terms.
    Invoked when the authors state that the penalized PDE is approximated using the Deep BSDE framework.
  • domain assumption The solution of the penalized PDE converges to the solution of the original free-boundary problem at a rate controlled by 1/λ.
    Standard assumption of penalty methods for variational inequalities; required for the O(1/λ) term to be meaningful.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs

    math.PR 2026-03 unverdicted novelty 6.0

    A two-grid penalization scheme for doubly reflected BSDEs achieves an explicit O(Δt^{1/2}) error bound when the penalty parameter λ is tuned as Δt^{-1/2} and the fine grid satisfies Δ̃t = O(Δt/λ²).

Reference graph

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