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Optimal stopping for a Wiener process with hidden Bernoulli drift is solved by lifting the problem to two dimensions via foliation, then slicing back.

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2026-06-26 06:49 UTC pith:NM5EUBDE

load-bearing objection The paper uses a foliation by an auxiliary displacement y to reduce the filtered optimal stopping problem to a family of one-dimensional interval problems whose boundaries are set by balancing conditions, then glues them into a two-dimensional region with coupled ODE boundaries.

arxiv 2606.23648 v1 pith:NM5EUBDE submitted 2026-06-22 math.PR math.OC

Optimal Stopping for a Diffusion with Unobserved Bernoulli Drift

Georgy Gaitsgori , Ioannis Karatzas This is my paper
classification math.PR math.OC
keywords optimal stoppingWiener processunobserved Bernoulli driftfilteringcontinuation regionfree boundaryfoliation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper reduces the original one-dimensional optimal stopping problem, after filtering the unobserved drift, to a Markovian problem whose solution requires an auxiliary parameter y that tracks displacement from the starting point. By solving the lifted two-dimensional problem for each fixed starting position x and then restricting to the slice y=0, the authors obtain an explicit description of the continuation region under structural assumptions on the terminal cost. The value function takes a semi-explicit form, the two-dimensional free boundaries obey monotonicity and a coupled ODE system, and the original problem admits a threshold solution whenever the y=0 slice intersects the continuation region. A sympathetic reader cares because the reduction turns an intractable filtering-plus-stopping problem into a concrete geometric construction whose boundaries can be computed or approximated directly.

Core claim

Under suitable structural assumptions on the terminal cost, each fixed-x continuation section is either empty or a single bounded interval whose endpoints are determined uniquely by a balancing condition; the value function is given in semi-explicit form, the two-dimensional continuation region is obtained by gluing, its free boundaries satisfy natural monotonicity and at regular points a coupled system of ODEs, and the original problem admits a threshold-type solution whenever the horizontal slice y=0 enters the two-dimensional continuation region.

What carries the argument

Foliation by the auxiliary displacement parameter y, which lifts the filtered one-dimensional problem to the plane so that fixed-x sections become intervals whose endpoints are fixed by a balancing condition before gluing produces the two-dimensional continuation region.

Load-bearing premise

The terminal cost must satisfy structural assumptions that force each fixed-x continuation section to be either empty or a single bounded interval with uniquely determined endpoints.

What would settle it

A concrete terminal cost obeying the stated structural assumptions for which the optimal continuation set at some fixed x consists of two disjoint intervals rather than one or none.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The two free boundaries of the two-dimensional continuation region are monotone and, at regular points, satisfy a coupled system of ordinary differential equations.
  • Whenever the slice y=0 lies inside the two-dimensional continuation region, the original one-dimensional problem has a threshold-type solution.
  • The value function of the lifted problem is given in semi-explicit form once the interval endpoints are known.
  • The original problem is recovered by restricting the two-dimensional solution to the plane y=0.

Where Pith is reading between the lines

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

  • The same foliation technique may apply to other optimal stopping problems with partial observations whose filtered state lives on the line but whose value depends on an auxiliary displacement variable.
  • Numerical solution of the coupled ODE system for the free boundaries would yield computable approximations to the optimal stopping set for concrete terminal costs.
  • If the structural assumptions on the terminal cost are dropped, the continuation sections may fragment into multiple intervals and the semi-explicit characterization would no longer hold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript solves an optimal stopping problem for a Wiener process with unobserved Bernoulli drift, subject to a symmetric terminal cost that increases with distance from the origin and a positive running cost c. After filtering, the problem is Markovian in the centered state x. The authors introduce an auxiliary parameter y representing displacement from initial position to foliate the problem, solve the augmented problem in the (x,y)-plane by characterizing fixed-x continuation sections as bounded intervals via balancing conditions, glue them to obtain the 2D continuation region whose free boundaries satisfy monotonicity and a coupled ODE system at regular points, and recover a threshold-type solution for the original problem on the y=0 slice under the structural assumptions on the terminal cost.

Significance. If the derivations hold, the paper offers a semi-explicit characterization of the value function and continuation region for this filtered optimal stopping problem, extending techniques like foliation and balancing conditions to this setting. This could be valuable for problems with partial observations in stochastic control, providing concrete structural results on the form of the solution. The direct derivation from filtered dynamics without fitted parameters is a strength.

minor comments (3)
  1. [Abstract] Abstract: the structural assumptions on the terminal cost are invoked to guarantee that fixed-x continuation sections are intervals with uniquely determined endpoints, but their precise form is not stated even at a high level; adding one sentence summarizing the key properties (e.g., convexity or growth conditions) would help readers evaluate the scope.
  2. The phrase 'semi-explicit form' for the value function is used repeatedly; specifying whether this means an integral representation, an explicit formula in terms of the boundaries, or a numerical ODE solution would clarify the degree of explicitness achieved.
  3. The description of the coupled ODE system for the free boundaries at regular points would benefit from a brief indication of the variables involved (e.g., which derivatives appear) to aid readability before the full derivation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and the positive assessment of its contributions. The referee's description accurately reflects the filtering approach, the foliation by the auxiliary parameter y, the characterization of the continuation region via balancing conditions and gluing, and the recovery of the threshold solution on the y=0 slice. The recommendation for minor revision is noted; however, the major comments section contains no specific points.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from dynamics and assumptions

full rationale

The paper derives the continuation region and value function via filtering to a Markov problem, foliation with auxiliary parameter y, section-wise solution of fixed-x intervals under explicit structural assumptions on the terminal cost (ensuring bounded intervals with unique balancing endpoints), gluing to a 2D region, and monotonicity/ODE characterization of boundaries. These steps follow directly from the stochastic dynamics and imposed assumptions without any quantity defined in terms of itself, without fitted parameters renamed as predictions, and without load-bearing self-citations or uniqueness theorems imported from prior author work. The resulting threshold solution on y=0 is a consequence of the construction rather than an input. This is the standard case of an internally consistent technical characterization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard stochastic calculus (Ito formula, filtering for a hidden Bernoulli drift) and on the structural assumptions on the terminal cost that are invoked to guarantee single-interval continuation sections. No free parameters are fitted to data; no new entities are postulated.

axioms (2)
  • standard math The filtered belief process together with the displacement coordinate y forms a Markov process whose infinitesimal generator is known explicitly from the Wiener dynamics and the Bernoulli prior.
    Invoked when the problem is reduced to Markovian optimal stopping after filtering (abstract, sentence beginning "After filtering...").
  • domain assumption The terminal cost is symmetric, increases with distance from the origin, and satisfies additional structural conditions that force each fixed-x continuation set to be empty or a single bounded interval.
    Stated explicitly as the condition under which the per-x problems admit the claimed interval geometry (abstract, paragraph beginning "Following this procedure...").

pith-pipeline@v0.9.1-grok · 5819 in / 1831 out tokens · 23935 ms · 2026-06-26T06:49:59.013196+00:00 · methodology

0 comments
read the original abstract

We solve fairly explicitly an optimal stopping problem for a Wiener process with unobserved Bernoulli drift, in the presence of a cost on terminal position which is symmetric and increases with distance from the origin, and of a fixed positive cost per unit time \(c > 0\). After filtering, the problem reduces to Markovian optimal stopping with complete observations for the state process ``centered'' by its starting position $x \in \mathbb R$. However, the solution becomes possible only after foliating by an additional state-parameter \(y \in \mathbb{R}\), representing the displacement from the initial position; this foliation ``lifts'' the problem from the real line to the plane, solves the augmented problem for each fixed initial position \(x\), characterizes fairly explicitly the optimal stopping region in \((x,y)\)-space, and finally obtains the solution of the original problem by ``slicing'' along \(y=0\). Following this procedure, we show that, under suitable structural assumptions on the terminal cost, each fixed-\(x\) continuation section is either empty or a single bounded interval, whose endpoints are determined uniquely by a balancing condition; the corresponding value function is then given in semi-explicit form. The two-dimensional continuation region is obtained by gluing these fixed-\(x\) intervals over \(x\); its two free boundaries satisfy natural monotonicity properties and, at regular points, can be described by a coupled system of ordinary differential equations. The resulting description yields a threshold-type solution of the original one-dimensional problem whenever the horizontal slice \(y=0\) enters the two-dimensional continuation region.

Figures

Figures reproduced from arXiv: 2606.23648 by Georgy Gaitsgori, Ioannis Karatzas.

Figure 1
Figure 1. Figure 1: The scale function and the supporting tangent line. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The one-valley geometry of Φx and the balancing level qx for k(s) = s 2 , c = 1, p = 0.3, x = 3. The shaded regions represent the positive and negative parts of the signed dG-area in Jx(qx). part remains across the valley. Hence lim q↓q x Jx(q) > 0. Similarly, as q ↑ qx , the positive part shrinks to zero, while a strictly negative part remains across the valley. Hence lim q↑qx Jx(q) < 0. By continuity, Jx… view at source ↗
Figure 3
Figure 3. Figure 3: The standard quadratic example k(s) = s 2 , with c = 0.6 and p = 0.5. The left panel shows the continuation strip. The right panel shows the surface Φ(x, y), several fixed-x sections y 7→ Φ(x, y), and the continuation strip projected onto the plane z = 0 [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two-dimensional continuation regions for two costs. The left panel shows quadratic cost [PITH_FULL_IMAGE:figures/full_fig_p041_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two exponential examples with different behaviors of the continuation set for the original [PITH_FULL_IMAGE:figures/full_fig_p042_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two examples with mixed power/exponential costs that fail to satisfy assumption [PITH_FULL_IMAGE:figures/full_fig_p042_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A numerical example outside the one-valley geometry. [PITH_FULL_IMAGE:figures/full_fig_p044_7.png] view at source ↗

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