arxiv: 2604.23608 · v2 · submitted 2026-04-26 · 💱 q-fin.TR · q-fin.GN· q-fin.ST

Recognition: unknown

Non-unique time and market incompleteness

Chris Angstmann, Tim Gebbie

Pith reviewed 2026-05-08 04:57 UTC · model grok-4.3

classification 💱 q-fin.TR q-fin.GNq-fin.ST

keywords market incompletenessevent-driven timepoint processesoperational timeno-arbitragerandom walksorder flowrenewal theory

0 comments

The pith

Non-unique continuum limits from event-driven time point to foundational market incompleteness.

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

Financial markets are typically modeled assuming a unique continuous time, but they operate asynchronously with event-driven order flow and random waiting times. By contrasting standard models with event-time, renewal, and point-process descriptions, the analysis shows that the continuum limit of a discrete-time random walk need not be unique. This non-uniqueness indicates a more foundational market incompleteness than usually considered. Such insights matter for understanding no-arbitrage conditions and risk-neutral pricing in realistic settings, while noting that effective completeness can appear at lower frequencies for practical risk management.

Core claim

In settings where the market is represented as a discrete event system and where the continuum limit of a discrete-time random walk need not be unique, the central suggestion is that such non-uniqueness points to a more foundational form of market incompleteness than is usually emphasized. This highlights the importance of operational time at the level of decision making but reminds that managing risk often requires reconciling operational time with a global calendar time, where forms of effective or average completeness may still emerge at lower frequencies.

What carries the argument

the non-uniqueness of the continuum limit of a discrete-time random walk in an event-driven point-process setting

If this is right

No-arbitrage, no-dynamic-arbitrage, and risk-neutral option pricing must be revisited in discrete event systems.
High-frequency hedging and execution expose mismatches between trading, pricing, and longer-horizon allocation clocks.
Effective or average completeness may still emerge at lower frequencies and remain useful for portfolio construction and risk management.

Where Pith is reading between the lines

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

Models may need to incorporate multiple possible continuum limits to capture incompleteness accurately in high-frequency event data.
This suggests new tests for inconsistencies in arbitrage opportunities across assets using different operational time scalings.
Hybrid strategies that switch between operational and calendar time based on trading frequency could improve risk management.

Load-bearing premise

The continuum limit of a discrete-time random walk in an event-driven point-process setting need not be unique, and this non-uniqueness produces a deeper incompleteness than standard formulations capture.

What would settle it

Demonstrating that different event-time scalings applied to the same asynchronous market data yield distinct sets of arbitrage-free prices or risk-neutral measures would support the claim; conversely, finding a unique effective limit regardless of scaling would challenge it.

read the original abstract

Financial markets are often modelled as if time were unique and continuous across assets and markets. Financial markets are however asynchronous, order flow is event-driven, and waiting times between events are often random. Many of the most influential formulations of financial market models presuppose a unique global calendar time and advocate for this or that preferred single latent continuous-time price system. Here we critically contrast these assumptions with event-time, renewal, point-process, and order-flow descriptions. We revisit no-arbitrage, no-dynamic-arbitrage, and risk-neutral option pricing in settings where the market is represented as a discrete event system and where the continuum limit of a discrete-time random walk need not be unique. The central suggestion is then that such non-uniqueness points to a more foundational form of market incompleteness than is usually emphasized. This highlights the importance of operational time at the level of decision making but reminds market practitioners that managing risk itself often requires reconciling operational time with a global calendar time. At these longer time scales forms of effective or average completeness may still emerge at lower frequencies and remain useful for portfolio construction and risk management, even if high-frequency hedging and execution expose a clock mismatch between trading, pricing, and longer-horizon allocation.

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.

Desk Editor's Note private letter to a colleague

The paper suggests non-unique continuum limits in event-driven markets create deeper incompleteness than standard models, but offers only a conceptual contrast without derivations or examples.

read the letter

The main thing to know is that the authors argue asynchronous, event-driven markets make the continuum limit from discrete steps non-unique, which they say creates a more basic form of incompleteness than the usual focus on missing traded assets or jumps. They contrast the common assumption of a single global calendar time with point-process and renewal descriptions where waiting times are random and order flow drives everything. This non-uniqueness, they suggest, affects no-arbitrage and risk-neutral pricing at a foundational level, while still allowing averaged completeness at longer horizons for portfolio work. The split between operational time for actual trading decisions and calendar time for risk management is the practical hook they emphasize, especially for high-frequency execution and multi-scale hedging. That part lands as a fair reminder of something practitioners already feel but models often gloss over. What the paper does reasonably is lay out the contrast cleanly and avoid overclaiming a full theorem. It revisits the relevant concepts without forcing a new mathematical object. The soft spot is that the central suggestion stays at the level of plausibility. The abstract gives no random-walk example with two different limits, no calculation of how attainable claims change, and no bounds on the resulting incompleteness. Without those steps it is difficult to tell whether this is genuinely deeper than existing treatments of asynchronous trading or just a reframing of known issues in jump and point-process models. The work draws on prior event-time literature but does not add new results or data. This is for readers already thinking about modeling foundations in quantitative finance, particularly those working on high-frequency or multi-horizon problems. Someone expecting theorems, simulations, or empirical checks will come away wanting more. I would send it for peer review. The topic is relevant and the suggestion could sharpen with concrete examples; referees can push on whether the incompleteness claim holds up once formalized.

Referee Report

2 major / 1 minor

Summary. The manuscript argues that standard financial models assume a unique continuous calendar time across assets, but real markets are asynchronous and event-driven with random waiting times. It contrasts this with event-time, renewal, point-process, and order-flow representations, revisiting no-arbitrage, no-dynamic-arbitrage, and risk-neutral pricing in discrete-event systems. The central suggestion is that the continuum limit of a discrete-time random walk in such settings need not be unique, implying a more foundational form of market incompleteness than usually emphasized, while noting that effective or average completeness may still emerge at lower frequencies for portfolio construction and risk management.

Significance. If the non-uniqueness argument can be formalized, the paper could usefully redirect attention to operational time scales in high-frequency settings and the practical reconciliation of event-driven clocks with global calendar time. It correctly identifies that managing risk often requires bridging these scales, which is a relevant observation for microstructure and execution. However, the conceptual framing without derivations, examples, or quantified consequences limits its immediate technical contribution to the literature on incompleteness.

major comments (2)

[Abstract] Abstract: The suggestion that non-uniqueness of the continuum limit 'points to a more foundational form of market incompleteness than is usually emphasized' is the load-bearing claim, yet no concrete model, theorem, or counter-example is supplied showing how this form exceeds standard incompleteness (e.g., from Poisson jumps or stochastic intensity). The argument therefore rests on plausibility rather than demonstrated distinction.
[Abstract] Abstract (paragraph on no-arbitrage and risk-neutral pricing): The revisit of these concepts in discrete-event systems is stated without any explicit conditions, measure-theoretic setup, or illustration of how non-unique limits alter arbitrage-free prices or risk-neutral measures. This absence prevents assessment of whether the claimed incompleteness is technically new or already captured by existing point-process frameworks.

minor comments (1)

[Abstract] Abstract: The phrase 'the continuum limit of a discrete-time random walk need not be unique' would benefit from a short parenthetical reference to a specific class of waiting-time distributions or point processes (e.g., renewal or Hawkes) to make the non-uniqueness claim more operational.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. The comments correctly identify that the paper is primarily conceptual in nature. We address each major comment below, indicating where revisions have been made to improve clarity and concreteness while preserving the intended scope of the work.

read point-by-point responses

Referee: [Abstract] Abstract: The suggestion that non-uniqueness of the continuum limit 'points to a more foundational form of market incompleteness than is usually emphasized' is the load-bearing claim, yet no concrete model, theorem, or counter-example is supplied showing how this form exceeds standard incompleteness (e.g., from Poisson jumps or stochastic intensity). The argument therefore rests on plausibility rather than demonstrated distinction.

Authors: We acknowledge that the central claim would be strengthened by a concrete illustration distinguishing the proposed incompleteness from standard sources such as jumps or stochastic intensities. The manuscript is positioned as a conceptual discussion of how non-unique operational time in asynchronous, event-driven markets can lead to non-unique continuum limits, which in turn implies a form of incompleteness rooted in the time scale itself rather than in the randomness of a fixed-time process. To address the referee's point directly, we have added a short illustrative example in the revised manuscript showing how two different renewal processes on the same event sequence can converge to distinct limiting diffusions, producing different no-arbitrage price intervals. This example is kept simple and does not constitute a full theorem, but it makes the distinction from existing point-process models more explicit. revision: yes
Referee: [Abstract] Abstract (paragraph on no-arbitrage and risk-neutral pricing): The revisit of these concepts in discrete-event systems is stated without any explicit conditions, measure-theoretic setup, or illustration of how non-unique limits alter arbitrage-free prices or risk-neutral measures. This absence prevents assessment of whether the claimed incompleteness is technically new or already captured by existing point-process frameworks.

Authors: We agree that the abstract's treatment of no-arbitrage and risk-neutral pricing is stated at a high level without explicit technical conditions. The full manuscript contrasts calendar-time and event-time representations and notes that the continuum limit need not be unique, but we accept that this leaves open the question of precise conditions and relation to existing frameworks. In the revision we have expanded the relevant discussion to include a brief outline of the discrete-event setup under which no-dynamic-arbitrage can be defined via predictable compensators, together with the observation that non-uniqueness of the limiting measure can generate a larger set of equivalent martingale measures than those arising from intensity uncertainty alone. This addition helps locate the argument relative to standard point-process models while keeping the paper's conceptual focus. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a conceptual critique contrasting continuous-time financial models with event-driven point-process and renewal descriptions of markets. It revisits no-arbitrage and risk-neutral pricing in discrete-event settings and suggests that non-uniqueness of continuum limits implies a deeper form of incompleteness. No equations, derivations, fitted parameters, or predictions appear in the provided text. The central claim is explicitly framed as a suggestion rather than a formal result obtained from self-referential steps. No self-citations, ansatzes, or uniqueness theorems are invoked in a load-bearing way. The argument draws on standard distinctions between calendar time and operational time without reducing any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The argument relies on the domain assumption that markets are asynchronous and event-driven, but this is not formalized in the provided text.

pith-pipeline@v0.9.0 · 5510 in / 1115 out tokens · 18092 ms · 2026-05-08T04:57:24.865703+00:00 · methodology

discussion (0)

Reference graph

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