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arxiv: 2606.23962 · v1 · pith:B4OJKNGFnew · submitted 2026-06-22 · 🧮 math.AP · math.ST· stat.TH

Delay-Penalty Comparison for Sequential Testing and Quickest Detection in State-Dependent Diffusion Models

Ye Liang This is my paper

Pith reviewed 2026-06-26 07:28 UTC · model grok-4.3

classification 🧮 math.AP math.STstat.TH
keywords sequential testingquickest detectiondiffusion observationsdelay penalty comparisonfree-boundary problemsposterior processstate-dependent driftShiryaev problem
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The pith

A pointwise larger running delay penalty increases the value of continuation, shrinks the continuation region, and produces earlier stopping in sequential testing and quickest detection for diffusions with state-dependent drift.

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

The paper formulates both sequential hypothesis testing and Bayesian quickest detection for observations from a diffusion whose drift switches between two alternatives, where the signal-to-noise ratio can depend on the current state. In this setting the posterior alone is not Markov, so the authors work with the two-dimensional process that augments the posterior (or likelihood ratio) with the observed diffusion itself. Within this common framework they prove a comparison result: when terminal costs are held fixed, raising the running delay penalty pointwise raises the value of waiting, contracts the continuation set, and advances the optimal stopping time. When the stopping set admits a one-sided representation in the posterior coordinate, the comparison supplies a monotonicity relation between the optimal alarm boundaries for different delay penalties.

Core claim

For a fixed terminal false-alarm or decision cost, any pointwise increase in the running delay penalty strictly increases the continuation value, strictly shrinks the continuation region, and therefore yields strictly earlier stopping. When the stopping set is one-sided in the posterior coordinate, the optimal alarm thresholds are ordered by the delay penalty. The result covers both linear delay costs and nonlinear marginal penalties after suitable Markovian augmentation of the state, and is verified numerically on a constant signal-to-noise Shiryaev problem in which the computed threshold rises monotonically with the delay cost.

What carries the argument

The augmented Markov process consisting of the posterior probability (or likelihood ratio) together with the observed diffusion, cast as a pair of degenerate free-boundary problems.

If this is right

  • Optimal stopping boundaries for different delay penalties are ordered when the stopping set is one-sided in the posterior.
  • The same comparison holds after Markovian augmentation for nonlinear marginal delay penalties.
  • The framework produces explicit monotone dependence of the alarm threshold on the delay cost in the constant signal-to-noise Shiryaev example.
  • Both sequential testing and quickest detection are handled by identical free-boundary problems once the state is augmented.

Where Pith is reading between the lines

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

  • The ordering result may extend to models with more than two drift alternatives if the posterior vector can be reduced to a scalar statistic with one-sided stopping sets.
  • Designers of monitoring systems could use the comparison to select the smallest delay penalty that still meets a prescribed average detection delay without recalculating the entire boundary.
  • The geometric effect of state-dependent information on the continuation region can be read off directly from the comparison without solving the free-boundary problem explicitly.

Load-bearing premise

The augmented posterior-diffusion process is a sufficient Markov state that reduces both testing and quickest detection to free-boundary problems with the same structure.

What would settle it

Compute the optimal alarm boundary numerically for two delay-penalty functions that differ by a positive constant on an interval; the boundary for the larger penalty must lie strictly below the boundary for the smaller penalty throughout that interval.

read the original abstract

We study sequential testing and Bayesian quickest detection for diffusion observations whose drift changes between two alternatives while the signal-to-noise ratio may depend on the current observation. In this setting the posterior probability is generally not a closed one-dimensional Markov statistic: the natural sufficient state is the augmented process consisting of the posterior (or likelihood ratio) and the observed diffusion. We formulate both testing and quickest detection within this common filtering framework and identify the corresponding degenerate free-boundary problems. The main contribution is a delay-penalty comparison principle. For a common terminal false-alarm or terminal decision cost, a pointwise larger running delay penalty increases the value of continuation, shrinks the continuation region, and yields earlier stopping. When the stopping set has a one-sided posterior representation, this gives an order relation for the optimal alarm boundaries. The result applies to linear delay costs and to nonlinear marginal delay penalties after the appropriate Markovian augmentation, and is illustrated by a constant signal-to-noise Shiryaev example in which the alarm threshold is computed numerically and shown to be monotone in the delay cost. The framework clarifies how state-dependent information and nonlinear delay costs jointly affect the geometry of sequential testing and quickest-detection rules.

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

0 major / 3 minor

Summary. The paper formulates sequential testing and Bayesian quickest detection for diffusions with state-dependent signal-to-noise ratios as optimal stopping problems on the augmented Markov state consisting of the posterior (or likelihood ratio) and the current observation. Both problems are cast as degenerate free-boundary problems. The central result is a delay-penalty comparison principle: for fixed terminal false-alarm/decision cost, a pointwise larger running delay penalty strictly increases the continuation value, strictly shrinks the continuation region, and produces earlier stopping; when the stopping set admits a one-sided posterior representation this yields an ordering of the optimal alarm boundaries. The result is stated for both linear costs and nonlinear marginal penalties (after Markovian augmentation) and is illustrated numerically on a constant-SNR Shiryaev problem showing monotone dependence of the threshold on the delay cost.

Significance. If the comparison principle is established rigorously, the result supplies a general, easily applicable monotonicity tool for sensitivity analysis of optimal boundaries with respect to delay penalties in filtering-based detection problems. The unified augmented-state framework handles both state-dependent information and nonlinear costs without requiring the posterior alone to be Markovian. The numerical Shiryaev example supplies concrete, falsifiable evidence of the predicted monotonicity. These features make the contribution potentially useful for both theoretical comparisons and practical tuning of detection rules.

minor comments (3)
  1. [§2–3] §2–3: the precise boundary conditions and growth assumptions needed to guarantee uniqueness of the value function for the degenerate free-boundary problems are stated only implicitly; an explicit list of the conditions used in the comparison argument would improve readability.
  2. The numerical illustration in the Shiryaev example reports monotonicity of the threshold but does not tabulate the computed values or the discretization parameters; adding a short table or convergence check would strengthen the supporting evidence.
  3. [Introduction] Notation for the augmented process (posterior + observation) is introduced without a dedicated symbol list; a brief table of symbols at the end of the introduction would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the paper's framework, the delay-penalty comparison principle, and its applicability to both linear and nonlinear costs in the augmented-state setting. Since no specific major comments were raised, we have no points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claim is a monotonicity result for optimal stopping: a pointwise larger running delay penalty raises the continuation value and shrinks the continuation region when terminal costs and the augmented Markov state (posterior plus observation) are held fixed. This is obtained from the standard comparison principle for optimal stopping problems and does not reduce to any fitted parameter, self-definition, or self-citation chain. The one-sided posterior representation is invoked only conditionally to translate the value-function ordering into a boundary ordering; the framework itself is stated to apply after Markovian augmentation for both linear and nonlinear costs. No equation or construction in the provided text equates a derived quantity to its own input by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5734 in / 1137 out tokens · 34286 ms · 2026-06-26T07:28:58.900242+00:00 · methodology

discussion (0)

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