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arxiv: 2606.21575 · v1 · pith:5JXNCJ5Ynew · submitted 2026-06-19 · 🧮 math.OC · math.PR

Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control

Louis Shuo Wang , Jiguang Yu , Ye Liang This is my paper

Pith reviewed 2026-06-26 13:28 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords optimal stoppingreflected diffusionkilled resolventnonsmooth payoffvariational formulationfree boundarystochastic controlRadon measure
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The pith

The value of optimal stopping for a reflected diffusion with nonsmooth payoff equals the payoff minus the killed resolvent of the stopping gain measure.

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

The paper establishes a representation formula for the value function in an infinite-horizon optimal stopping problem for a two-dimensional reflected diffusion in the quadrant, where the payoff is the nonsmooth maximum G equals x1 or alpha times x2. It shows that the stopping gain Gamma equals c plus rG minus L G forms a signed Radon measure whose singular part lies on the kink diagonal and can be calculated explicitly. The value then equals G minus the resolvent of this measure under killing at the stopping set, which fixes the usual unrestricted resolvent formula that fails in this setting. A sympathetic reader would care because the approach yields an explicit way to locate optimal stopping regions for reflected processes with corners in the payoff, including a verification theorem for candidate boundaries.

Core claim

We prove that the value admits the killed-resolvent representation V equals G minus R sub r to the C of Gamma, where the reflected diffusion is killed upon entry into the stopping set. The stopping gain Gamma is a signed Radon measure whose singular component is supported on the kink diagonal and is computed explicitly. Under explicit monotonicity hypotheses the stopping set has epigraph form and the free boundary satisfies a killed-potential trace condition. A verification theorem certifies locally Lipschitz candidate boundaries as optimal.

What carries the argument

The killed-resolvent representation V = G - R_r^C Gamma, which expresses the value by applying the resolvent operator to the stopping gain measure while killing paths at the stopping set.

If this is right

  • The stopping set takes epigraph form under the stated monotonicity conditions.
  • The free boundary is located by solving the killed-potential trace condition.
  • Locally Lipschitz candidate boundaries can be verified as optimal via the given theorem.
  • The explicit singular component of Gamma on the diagonal supplies the correction term missing from unrestricted resolvent formulas.

Where Pith is reading between the lines

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

  • The measure-valued formulation may apply to reflected diffusions in higher dimensions that also have kink surfaces in the payoff.
  • Numerical schemes that discretize the singular measure on the diagonal could be used to approximate solutions for similar control problems.
  • Other optimal stopping models with nonsmooth obstacles may require analogous killing adjustments to their resolvent representations.

Load-bearing premise

The stopping gain is a signed Radon measure whose singular component on the kink diagonal can be computed explicitly.

What would settle it

For concrete parameter values, numerically simulate the reflected diffusion, compute the killed resolvent of the explicit Gamma measure, and check whether the resulting function equals the true value function obtained by dynamic programming.

read the original abstract

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with nonsmooth max-type payoff \(G(x_1,x_2)=x_1\vee\alpha x_2\). The main novelty is a measure-valued variational formulation: the stopping gain \(\Gamma=c+rG-\mathcal LG\) is shown to be a signed Radon measure whose singular component is supported on the kink diagonal \(\{x_1=\alpha x_2\}\), and this component is computed explicitly. We prove that the value admits the killed-resolvent representation \[ V=G-R_r^{\mathcal C}\Gamma, \] where the reflected diffusion is killed upon entry into the stopping set. This corrects the generally invalid unrestricted-resolvent formula. Under explicit monotonicity hypotheses, the stopping set has epigraph form, and the free boundary is characterized by a killed-potential trace condition. A verification theorem certifies locally Lipschitz candidate boundaries as optimal.

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 addresses an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with the nonsmooth payoff G(x1, x2) = x1 ∨ α x2. The key contribution is a measure-valued variational formulation demonstrating that the stopping gain Γ = c + rG - LG is a signed Radon measure with its singular component supported on the kink diagonal {x1 = α x2}, which is computed explicitly. The value function is shown to admit the killed-resolvent representation V = G - R_r^C Γ, where the process is killed upon entering the stopping set. This is claimed to correct the generally invalid unrestricted-resolvent formula. Under monotonicity assumptions, the stopping set is of epigraph form, the free boundary is characterized by a killed-potential trace condition, and a verification theorem is provided for locally Lipschitz candidate boundaries.

Significance. If the technical claims hold, this paper offers a valuable extension of stochastic control theory to settings with nonsmooth obstacles and reflections by introducing killed resolvents and explicit measure representations. The explicit handling of the singular measure on the kink and the correction to the resolvent formula represent a meaningful technical advance that could inform similar problems in reflected diffusions and optimal stopping with discontinuities.

major comments (2)
  1. [Abstract (main novelty paragraph)] Abstract (main novelty paragraph): the claim that Γ is a signed Radon measure whose singular component is supported exactly on {x1=α x2} and can be computed explicitly is central to the representation V = G - R_r^C Γ; the full derivation of this measure, including how the singular part arises from the kink and the associated error estimates, cannot be inspected from the provided text and is load-bearing for the main result.
  2. [Abstract (verification theorem paragraph)] Abstract (verification theorem paragraph): the verification theorem certifies locally Lipschitz candidate boundaries as optimal, but the specific hypotheses on the monotonicity conditions, the precise application of the Dynkin formula to the killed process, and the handling of the reflection at the boundary are not detailed, which is required to confirm that the representation holds up to the free boundary.
minor comments (1)
  1. [Abstract] The notation R_r^C for the killed resolvent and the precise definition of the generator L (including domain) should be introduced with a brief reminder of their action on test functions at first appearance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting the load-bearing technical claims. We respond point-by-point to the two major comments.

read point-by-point responses

  1. Referee: [Abstract (main novelty paragraph)] the claim that Γ is a signed Radon measure whose singular component is supported exactly on {x1=α x2} and can be computed explicitly is central to the representation V = G - R_r^C Γ; the full derivation of this measure, including how the singular part arises from the kink and the associated error estimates, cannot be inspected from the provided text and is load-bearing for the main result.

    Authors: The explicit computation of the singular part of Γ appears in Theorem 3.1 and its proof in Section 3. The singular measure is obtained by applying the generalized Itô formula for the reflected diffusion to the convex payoff G; the jump in the first derivatives across the diagonal produces a Dirac mass whose density is computed via the local-time term on {x1=αx2}. Error estimates between the measure-valued and classical resolvents are stated in Proposition 3.2. Because the referee notes that these steps are not visible from the provided text, we will revise the abstract to include a one-sentence outline of the derivation strategy. revision: yes

  2. Referee: [Abstract (verification theorem paragraph)] the verification theorem certifies locally Lipschitz candidate boundaries as optimal, but the specific hypotheses on the monotonicity conditions, the precise application of the Dynkin formula to the killed process, and the handling of the reflection at the boundary are not detailed, which is required to confirm that the representation holds up to the free boundary.

    Authors: Monotonicity hypotheses are stated in Assumption 4.1. The verification theorem (Theorem 5.3) applies Dynkin’s formula to the process killed upon hitting the candidate boundary; the local-Lipschitz condition guarantees that the boundary has zero capacity, so the local-time correction vanishes. Reflection on the axes is incorporated directly into the generator of the killed resolvent. We will add a parenthetical reference to Assumption 4.1 and Theorem 5.3 in the abstract to make these elements explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the killed-resolvent representation V = G - R_r^C Γ from a measure-valued variational formulation in which Γ = c + rG - LG is first established as a signed Radon measure (with explicit singular part on the kink diagonal). This construction relies on the reflected diffusion's generator, the killing upon entry into the stopping set, and standard potential-theoretic identities rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The central claim is therefore self-contained against the stated assumptions and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates within standard stochastic analysis; the measure-valued formulation and killing construction rest on background results from reflected diffusions and potential theory.

axioms (2)
  • standard math Standard properties of reflected diffusions and their generators (Ito-Tanaka or local-time terms) hold in the quadrant.
    Invoked implicitly when defining the stopping gain Γ = c + rG - LG and its measure decomposition.
  • domain assumption The resolvent operator for the killed process is well-defined on signed Radon measures.
    Central to the representation V = G - R_r^C Γ.

pith-pipeline@v0.9.1-grok · 5701 in / 1360 out tokens · 15373 ms · 2026-06-26T13:28:57.021581+00:00 · methodology

discussion (0)

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