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arxiv: 2606.27573 · v1 · pith:X2HXTT7Qnew · submitted 2026-06-25 · 🧮 math.PR

Conditional Path Decomposition at the Infimum and Maximum Drawdowns for Spectrally Negative L\'{e}vy Processes

Ceren Vardar Acar , Mine \c{C}a\u{g}lar This is my paper

Pith reviewed 2026-06-29 00:32 UTC · model grok-4.3

classification 🧮 math.PR
keywords spectrally negative Lévy processespath decompositionmaximum drawdownscale functionsDoob h-transforminfimumexponential timefluctuation theory
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The pith

For spectrally negative Lévy processes observed up to an independent exponential time, pre-infimum paths from two decompositions at the infimum and supremum are characterized by scale functions as Doob h-transforms of killed processes, yiel

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

The paper derives scale-function expressions for the law of the path segment before the process reaches its infimum, first under a simple decomposition into pre-infimum and post-infimum pieces, and second under an ordering where the infimum occurs before the supremum, splitting the path into three independent segments. These conditional laws appear as Doob h-transforms of the original process killed at an independent exponential time. The transforms produce closed-form distributions for the maximum drawdown attained on each segment. The same machinery recovers the classical path decompositions when the driving process is Brownian motion.

Core claim

The conditional law of the pre-infimum component is a Doob h-transform of the killed spectrally negative Lévy process whose h-function is built from the scale functions; under the additional ordering the intermediate and post-supremum segments receive analogous transforms; these laws determine the exact distribution of the maximum drawdown on every independent piece and the supremum reached by the pre-infimum piece.

What carries the argument

Scale functions of the spectrally negative Lévy process, used to construct the Doob h-transforms that encode the conditioned pre-infimum, intermediate, and post-supremum path laws.

If this is right

  • The maximum drawdown on the pre-infimum path admits an explicit formula in terms of scale functions.
  • The supremum attained by the pre-infimum process has an explicit distribution derived from the same scale functions.
  • The intermediate and post-supremum components each carry independent maximum-drawdown distributions given by the corresponding h-transforms.
  • The full set of decompositions reduces exactly to the classical Brownian-motion identities when the Lévy process has no jumps.

Where Pith is reading between the lines

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

  • The same scale-function expressions could be used to compute the joint law of drawdown and the time spent below a level on each path segment.
  • In insurance models the formulas supply exact ruin probabilities conditional on the timing of the global minimum.
  • Numerical inversion of the Laplace transforms implicit in the scale functions would allow simulation of conditioned paths without discretizing the Lévy measure.

Load-bearing premise

The underlying process must be spectrally negative so that its scale functions exist and fully characterize the exit problems needed for the h-transforms.

What would settle it

A direct calculation, for Brownian motion with drift, of the maximum drawdown distribution on the pre-infimum segment that fails to match the known classical formula obtained from the same decomposition.

read the original abstract

We study maximum-drawdown laws conditioned on extremes for a spectrally negative L\'evy process and observed up to an independent exponential time. The main contribution is a set of scale-function characterizations of the pre-infimum path arising from two decompositions of the process. The first is the decomposition at the infimum into pre-infimum and post-infimum components. The second, under the ordering in which the infimum is attained before the supremum, decomposes the path into pre-infimum, intermediate, and post-supremum components. We also identify the distribution of the supremum for the pre-infimum process in the first decomposition. The resulting conditional laws are expressed as Doob $h$-transforms of killed spectrally negative L\'evy processes and they yield explicit formulas for the maximum drawdown on each independent path component. The results confirm the classical decompositions for Brownian motion.

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 / 2 minor

Summary. The paper studies maximum-drawdown laws conditioned on extremes for a spectrally negative Lévy process observed up to an independent exponential time. It provides scale-function characterizations of the pre-infimum path from two decompositions (at the infimum, and under the ordering where infimum precedes supremum), expresses the conditional laws as Doob h-transforms of killed processes, derives explicit formulas for the maximum drawdown on each component, identifies the distribution of the supremum for the pre-infimum process, and verifies that the formulas recover the known Brownian-motion decompositions.

Significance. If the derivations hold, the work supplies explicit, scale-function-based expressions for conditional path laws and drawdown distributions that extend classical Brownian-motion results to spectrally negative Lévy processes. The reliance on established scale functions for exit problems and standard Doob h-transform constructions under exponential killing, together with the explicit recovery of the Brownian-motion case as a sanity check, strengthens the contribution for applications in risk theory and fluctuation theory.

minor comments (2)
  1. [Abstract / Introduction] The abstract and introduction would benefit from a brief statement of the precise form of the two decompositions (pre-/post-infimum and the three-component ordering) before the scale-function expressions are introduced, to improve readability for readers unfamiliar with the path-decomposition literature.
  2. [Section 2 / Preliminaries] Notation for the h-transforms and the associated killed processes should be introduced with a short reminder of the underlying scale function W and its derivative, even if standard, to make the transition from the classical exit formulas to the conditional laws fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have nothing further to address point by point. Any minor editorial or typographical issues will be corrected in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on the classical existence and exit-problem properties of scale functions for spectrally negative Lévy processes (standard external theory) together with the construction of Doob h-transforms under exponential killing. These are invoked as known inputs rather than derived within the paper. The explicit recovery of the known Brownian-motion decompositions supplies an independent external sanity check. No equation reduces a fitted parameter or self-citation to the target result by construction, and no load-bearing premise collapses to a self-referential definition or ansatz smuggled via the authors' prior work. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on the established theory of spectrally negative Lévy processes and their scale functions without introducing new free parameters or entities.

axioms (2)
  • domain assumption Spectrally negative Lévy processes admit scale functions that characterize their exit problems
    Central to all characterizations described in the abstract.
  • domain assumption Observation occurs up to an independent exponential time
    Enables the conditioning and memoryless properties used for the decompositions.

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Reference graph

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