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arxiv: 2606.17545 · v1 · pith:CHFNGIILnew · submitted 2026-06-16 · 💻 cs.LG · q-fin.CP· q-fin.PR

Continuous-time Optimal Stopping through Deep Reinforcement Learning

Cosmin Borsa , Michael Ludkovski This is my paper

Pith reviewed 2026-06-27 02:31 UTC · model grok-4.3

classification 💻 cs.LG q-fin.CPq-fin.PR
keywords optimal stoppingreinforcement learningdeep neural networksAmerican optionsBermudan optionscontinuous timeexercise boundaryadaptive sampling
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The pith

CARLOS uses an aggregate deep neural network to learn continuous-time optimal stopping boundaries by progressively refining grids and adaptive sampling.

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

Simulation solvers for optimal stopping must discretize time, but coarse grids undervalue the expected reward while fine grids let approximation errors accumulate in backward recursion. The CARLOS algorithm trains a single aggregate deep neural network across increasing time resolutions, beginning with coarse grids and raising stopping frequency in parallel with network training. An adaptive sampling strategy concentrates effort near the stopping boundary to improve efficiency. Benchmarked tests show CARLOS returns higher values than standard Bermudan solvers and nears the American upper bound while remaining computationally efficient relative to non-RL methods.

Core claim

CARLOS (Continuous-time Adaptive Reinforcement Learning for Optimal Stopping) utilizes an aggregate deep neural network (ADNN) to learn a joint space-time decision boundary. Starting from a coarse time grid, the frequency of stopping opportunities is progressively increased while training the ADNN in parallel to refine timing-value estimates, combined with an adaptive sampling strategy that concentrates effort near the stopping boundary.

What carries the argument

Aggregate Deep Neural Network (ADNN) that represents the joint space-time decision boundary for the exercise rule.

If this is right

  • CARLOS produces higher option prices than existing Bermudan solvers.
  • The values approach the American upper bound more closely than standard methods.
  • Computational efficiency remains high relative to non-RL comparators.
  • The exercise rule can be learned at arbitrarily fine time resolution.

Where Pith is reading between the lines

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

  • The joint space-time network representation may transfer to other stochastic control tasks that currently rely on fixed time discretizations.
  • Adaptive sampling near decision boundaries could reduce sample needs in related reinforcement learning problems with sparse rewards.
  • Direct implementation in existing option pricing software would allow side-by-side testing on market-calibrated models.

Load-bearing premise

The method assumes that progressively refining the time grid while training the aggregate deep neural network on the joint space-time boundary will converge to the true continuous-time optimum without bias from adaptive sampling or network approximation.

What would settle it

Running CARLOS on a problem with a known closed-form continuous-time optimum and checking whether the computed value exceeds all Bermudan discretizations yet remains strictly below the known true value without overshooting.

Figures

Figures reproduced from arXiv: 2606.17545 by Cosmin Borsa, Michael Ludkovski.

Figure 1
Figure 1. Figure 1: Optimal stopping boundaries of the 1-dimensional Bermudan Put option [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left Panel: Average rewards υ [ℓ] = Ave(Υ[ℓ] ) (left y-axis) and learning rates η [ℓ] (right y￾axis, log scale) across RL loops ℓ and exercise grids T (ex,b) for the B1 option from [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pricing the M2 contract from [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Input sets (t, x) [ℓ] for the B1 contract using the parameter configuration in [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
read the original abstract

Simulation based solvers for optimal stopping problems must discretize the stopping decision. Under classical dynamic programming, a coarse exercise grid with only a few stopping opportunities can materially undervalue the optimal expected reward, whereas on a very fine grid, approximation errors accumulate through the backward recursion. To remove this limitation, we develop a new reinforcement-learning inspired algorithm that enables us to learn the exercise rule at arbitrarily fine time resolution. Our CARLOS (Continuous-time Adaptive Reinforcement Learning for Optimal Stopping) algorithm utilizes an aggregate deep neural network (ADNN) to learn a joint space-time decision boundary. Starting from a coarse time grid, we progressively increase the frequency of stopping opportunities, while in parallel training the ADNN to refine its timing-value estimates. We moreover design an adaptive sampling strategy that gradually concentrates training effort near the stopping boundary. Benchmarked results show that CARLOS delivers higher prices than existing Bermudan solvers, approaching the American upper bound, and achieves high computational efficiency relative to non-RL comparators.

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

1 major / 0 minor

Summary. The paper introduces the CARLOS algorithm for continuous-time optimal stopping problems. It employs an aggregate deep neural network (ADNN) to learn a joint space-time decision boundary, beginning with a coarse exercise grid that is progressively refined while the network is trained in parallel and training samples are adaptively concentrated near the estimated stopping boundary. The central claim is that this procedure produces higher prices than standard Bermudan dynamic-programming solvers, approaches the American upper bound, and does so with improved computational efficiency.

Significance. If the convergence claim holds, the work would remove a long-standing discretization bias in simulation-based optimal stopping and supply a scalable RL route to high-resolution continuous-time problems. The adaptive-sampling and joint space-time network ideas are technically interesting and could be reused in other free-boundary problems.

major comments (1)
  1. The headline claim that CARLOS converges to the true continuous-time optimum (and therefore delivers prices approaching the American upper bound) rests on the unproven assertion that iterative grid refinement plus boundary-focused adaptive sampling drives both discretization and approximation error to zero. No error bounds, contraction argument, or continuous-time limit theorem are supplied to guarantee that the learned stopping set converges to the Snell envelope as mesh size → 0. This is load-bearing for the benchmark superiority statement.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the constructive comment. We respond to the major point below.

read point-by-point responses

  1. Referee: The headline claim that CARLOS converges to the true continuous-time optimum (and therefore delivers prices approaching the American upper bound) rests on the unproven assertion that iterative grid refinement plus boundary-focused adaptive sampling drives both discretization and approximation error to zero. No error bounds, contraction argument, or continuous-time limit theorem are supplied to guarantee that the learned stopping set converges to the Snell envelope as mesh size → 0. This is load-bearing for the benchmark superiority statement.

    Authors: We agree that the manuscript supplies no error bounds, contraction mapping, or continuous-time limit theorem establishing convergence of the learned stopping set to the Snell envelope. All superiority statements rest on numerical experiments in which CARLOS produces higher values than standard Bermudan dynamic-programming solvers and approaches the American upper bound. We will revise the abstract, introduction, and conclusion to state explicitly that the results are empirical, to remove any implication of proven convergence, and to add a dedicated limitations paragraph noting the absence of theoretical guarantees together with directions for future analysis. revision: yes

standing simulated objections not resolved
  • Supplying a rigorous convergence theorem, contraction argument, or error bounds that guarantee the learned stopping set converges to the Snell envelope as the mesh size tends to zero.

Circularity Check

0 steps flagged

No circularity; algorithmic procedure is independent of its inputs

full rationale

The CARLOS algorithm is presented as a standalone RL procedure that starts from a coarse grid, progressively refines the time discretization, trains an ADNN on the joint space-time boundary, and applies adaptive sampling near the boundary. No equations, fitted parameters, or self-citations are described that would make any claimed performance (higher prices approaching the American bound) equivalent to the inputs by construction. Benchmarking results are external to the method itself, and the derivation chain contains no self-definitional steps, fitted-input predictions, or load-bearing self-citations. The approach is self-contained against external comparators.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 2 invented entities

Abstract-only review yields no concrete free parameters, axioms, or invented entities beyond the high-level algorithmic components named; the central claim rests on the unverified effectiveness of the described neural-network training procedure.

invented entities (2)
  • CARLOS algorithm no independent evidence
    purpose: Enable learning of exercise rules at arbitrarily fine time resolution
    New method introduced in the abstract
  • Aggregate deep neural network (ADNN) no independent evidence
    purpose: Learn joint space-time decision boundary
    Core modeling component described in the abstract

pith-pipeline@v0.9.1-grok · 5699 in / 1129 out tokens · 55699 ms · 2026-06-27T02:31:35.024722+00:00 · methodology

discussion (0)

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