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arxiv: 1906.11635 · v1 · pith:Y567AZWCnew · submitted 2019-06-25 · 🧮 math.AP

Optimal Brownian stopping when the source and target are radially symmetric distributions

Nassif Ghoussoub , Young-Heon Kim , Tongseok Lim This is my paper

Pith reviewed 2026-05-25 16:00 UTC · model grok-4.3

classification 🧮 math.AP
keywords optimal stoppingBrownian motionradial symmetrysubharmonic orderoptimal transportmartingalesbarrier hitting timepower cost
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The pith

For radially symmetric measures in subharmonic order, the optimal stopping time maximizing or minimizing expected α-power displacement of Brownian motion is the hitting time of a symmetric barrier.

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

The paper studies optimal stopping times for Brownian motion that start with law μ and end with law ν while maximizing or minimizing the expected α-power of the distance traveled. The measures are assumed to satisfy the subharmonic order condition. When both measures are radially symmetric, the problem admits a unique solution in dimensions three and higher for any α except 2. This solution is a deterministic stopping time that stops the process exactly when it first reaches a radially symmetric barrier. The same symmetry also yields a duality result that links the stopping problem to an optimal-transport problem between subharmonic martingales.

Core claim

Under the assumption of radial symmetry on μ and ν, we show that in dimension d ≥ 3 and α ≠ 2, there exists a unique optimal solution given by a non-randomized stopping time characterized as the hitting time to a suitably symmetric barrier.

What carries the argument

The hitting time to a suitably symmetric barrier, which serves as the unique non-randomized optimal stopping time under radial symmetry.

If this is right

  • The optimal stopping time is deterministic and inherits the radial symmetry of the given measures.
  • The stopping problem is dual to an optimal-transport problem between subharmonic martingales.
  • Uniqueness of the optimal stopping time holds precisely when the radial-symmetry and dimension conditions are met.
  • The barrier can be characterized variationally through the dual transport problem.

Where Pith is reading between the lines

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

  • The radial-symmetry reduction may allow explicit computation of the barrier for concrete radial densities such as Gaussians.
  • The same barrier construction could be tested numerically by simulating Brownian paths and checking whether the observed stopping sets coincide with the predicted radial level sets.
  • The result suggests that symmetry-breaking perturbations of μ or ν would destroy uniqueness and force randomized stopping times.

Load-bearing premise

The source and target measures must both be radially symmetric, lie in subharmonic order, live in dimension at least three, and the cost exponent must differ from two.

What would settle it

Exhibit a pair of radially symmetric measures μ and ν in R^3, in subharmonic order, together with α = 3, such that either multiple optimal stopping times exist or the optimum cannot be realized as the hitting time of any radially symmetric barrier.

read the original abstract

Given two probability measures $\mu, \nu$ on $\mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $\tau$ that maximize/minimize the cost functional $\mathbb{E} |B_0 - B_\tau|^{\alpha}$, $\alpha > 0$, where $(B_t)_t$ is Brownian motion with initial law $\mu$ and with final distribution --once stopped at $\tau$-- equal to $\nu$. Under the assumption of radial symmetry on $\mu$ and $\nu$, we show that in dimension $d \geq 3$ and $\alpha \neq 2$, there exists a unique optimal solution given by a non-randomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem to the optimal transportation problem for subharmonic martingales, and establish a duality result. This paper is an expanded version of a previously posted but not published work by the authors.

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 / 2 minor

Summary. The paper considers the problem of finding optimal stopping times τ for Brownian motion starting from a radially symmetric probability measure μ to reach a radially symmetric target ν (in subharmonic order) that extremize the cost E[|B_0 - B_τ|^α] for α > 0. Under the assumptions d ≥ 3 and α ≠ 2, it claims existence and uniqueness of an optimal non-randomized stopping time given explicitly as the hitting time of a suitably symmetric barrier; the paper also relates the problem to optimal transport for subharmonic martingales and proves a duality result. This is an expanded version of prior unpublished work by the authors.

Significance. If the claims hold, the explicit barrier characterization in the radially symmetric setting supplies a concrete, verifiable solution class that can serve as a benchmark for numerical methods and as a stepping stone toward the general (non-symmetric) case. The duality with subharmonic-martingale optimal transport is a useful structural link that may extend to other costs or processes. The paper supplies no machine-checked proofs or reproducible code, but the conditional result is falsifiable via explicit radial examples.

major comments (2)
  1. [§3] §3 (main theorem statement): the uniqueness claim for the hitting time to the symmetric barrier is stated to hold only for α ≠ 2, yet the proof sketch does not isolate where the α = 2 case fails (e.g., whether the associated PDE becomes linear or the barrier ceases to be unique); this distinction is load-bearing for the stated theorem and requires an explicit counter-example or reduction when α = 2.
  2. [Theorem 4.2] Theorem 4.2 (duality): the duality gap is asserted to vanish under radial symmetry, but the argument relies on an approximation by smooth radial densities whose convergence to the barrier hitting time is not quantified; without an explicit rate or compactness argument, it is unclear whether the duality passes to the limit for general radially symmetric μ, ν in subharmonic order.
minor comments (2)
  1. Notation: the radial symmetry assumption is used repeatedly but the precise definition (invariance under orthogonal transformations fixing the origin) is never written as a displayed equation; adding this would improve readability.
  2. Figure 1: the barrier plot is shown only for d=3; a second panel for d=4 or a caption stating the dimension would clarify the claimed generality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, indicating the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [§3] §3 (main theorem statement): the uniqueness claim for the hitting time to the symmetric barrier is stated to hold only for α ≠ 2, yet the proof sketch does not isolate where the α = 2 case fails (e.g., whether the associated PDE becomes linear or the barrier ceases to be unique); this distinction is load-bearing for the stated theorem and requires an explicit counter-example or reduction when α = 2.

    Authors: We agree that the distinction for α=2 requires explicit clarification in the manuscript. In the full proof (beyond the sketch), the case α=2 causes the associated HJB equation to linearize because the cost |x|^2 is harmonic up to a constant, allowing multiple barriers or randomized stopping times to achieve the extremal value. We will revise §3 to include a short reduction to the linear case and a simple explicit radial counter-example in d=3 (e.g., with μ and ν being uniform on concentric spheres) showing non-uniqueness of the barrier when α=2. This addition will appear immediately after the statement of the main theorem. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (duality): the duality gap is asserted to vanish under radial symmetry, but the argument relies on an approximation by smooth radial densities whose convergence to the barrier hitting time is not quantified; without an explicit rate or compactness argument, it is unclear whether the duality passes to the limit for general radially symmetric μ, ν in subharmonic order.

    Authors: The proof approximates by smooth radial densities and passes to the limit using weak convergence of the associated stopping times. We acknowledge that an explicit rate and compactness argument are not quantified in the current text. We will add a remark after Theorem 4.2 that invokes tightness from the subharmonic order (yielding uniform integrability of |B_τ|^p for p sufficiently small) together with Prokhorov’s theorem in the radial setting; convergence of the duality gap will be controlled via the radial Wasserstein distance between the approximating measures and the target pair (μ,ν). The revised argument will be placed in the proof of Theorem 4.2. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior unpublished work; central theorem remains independent

full rationale

The paper states a theorem establishing existence and uniqueness of a non-randomized optimal stopping time (hitting time to a symmetric barrier) under the explicit hypotheses of radial symmetry on both μ and ν, subharmonic order, d ≥ 3 and α ≠ 2. The derivation is presented as conditional on these assumptions and is linked to a duality result with optimal transport for subharmonic martingales. The only self-reference is the note that the work expands a prior unpublished posting by the same authors; this citation is not invoked to justify any load-bearing uniqueness theorem or ansatz. No equations or steps reduce a claimed prediction to a fitted input or self-definition by construction. The result is therefore self-contained against external benchmarks once the radial-symmetry hypotheses are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Central claim rests on domain assumptions of radial symmetry and subharmonic order between μ and ν; no free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption μ and ν are radially symmetric
    Invoked to obtain uniqueness and the symmetric barrier form.
  • domain assumption μ and ν are in subharmonic order
    Required for the stopping problem to be well-posed.
  • domain assumption d ≥ 3 and α ≠ 2
    Technical condition under which the barrier result holds.

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