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arxiv: 2605.05904 · v1 · submitted 2026-05-07 · 🧮 math.PR

Recognition: unknown

Schr\"odinger's problem with constraints

Beatrice Acciaio, Umut \c{C}et\.in

Pith reviewed 2026-05-08 06:26 UTC · model grok-4.3

classification 🧮 math.PR
keywords Schrödinger problemBrownian bridgeKyle modeltrading costsdefault riskfinancial equilibriumconstrained bridge
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The pith

A broad class of constrained bridges solve Schrödinger-type problems, implying that equilibria with trading costs converge to the classical Kyle model.

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

The paper begins with the established link between the Kyle equilibrium under a static private signal and the Brownian bridge. It extends this to a larger family of bridges capable of encoding features such as trading costs and default risk inside equilibrium models. These generalized bridges are proven to solve Schrödinger problems that incorporate the same constraints. The connection then delivers a convergence result: equilibria arising in models with positive trading costs approach the equilibria of the classical Kyle model as those costs tend to zero. A sympathetic reader would care because the result supplies a single probabilistic mechanism for studying Kyle-style equilibria under realistic market frictions.

Core claim

Motivated by the connection between the Kyle equilibrium with static private signal and the Brownian bridge, we study a much broader class of bridges that allow one to consider more general equilibrium models, for example ones including trading costs and default risk. We show that such bridges are solutions to problems of the Schrödinger-type. Leveraging this connection, we obtain that the equilibria in models with trading costs converge to equilibria in the classical Kyle model.

What carries the argument

The constrained bridge, defined to satisfy the equilibrium conditions under trading costs or default risk, which is shown to solve a Schrödinger-type variational problem.

If this is right

  • Equilibria in models with trading costs exist and are given by constrained bridges.
  • As trading costs approach zero, the associated equilibria recover the classical Kyle equilibria.
  • Default risk can be incorporated by choosing appropriate constraints on the same bridge class.
  • Tools developed for Schrödinger problems can be applied directly to compute or characterize these financial equilibria.

Where Pith is reading between the lines

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

  • The same representation may accommodate additional market constraints such as position limits or asymmetric information.
  • Numerical methods for solving Schrödinger problems could yield approximate equilibria for Kyle models that include small frictions.
  • The convergence result suggests that the classical Kyle model remains a good approximation when trading costs are modest.

Load-bearing premise

Generalized equilibrium models with trading costs and default risk can be represented as constrained bridge problems to which the Schrödinger connection extends rigorously.

What would settle it

Exhibit a concrete equilibrium model with positive trading costs whose equilibrium prices or strategies fail to approach the classical Kyle equilibrium as costs are driven to zero.

read the original abstract

Motivated by the connection between the Kyle equilibrium with static private signal and the Brownian bridge, we study a much broader class of bridges that allow one to consider more general equilibrium models, for example ones including trading costs and default risk. We show that such bridges are solutions to problems of the Schr\"odinger-type. Leveraging this connection, we obtain that the equilibria in models with trading costs converge to equilibria in the classical Kyle model.

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 motivates a broad class of constrained bridges from generalized Kyle-type equilibria that incorporate trading costs and default risk. It claims these bridges solve constrained Schrödinger problems, and uses the connection to deduce that equilibria with positive trading costs converge to the classical Kyle equilibrium (with static private signal and Brownian bridge) as costs vanish.

Significance. If the technical steps hold, the work supplies a unified optimal-transport perspective on equilibrium models with frictions, extending the known Brownian-bridge characterization of the Kyle model. The convergence result, if rigorously established, would be a concrete contribution to mathematical finance, allowing systematic approximation of frictionless equilibria by models with costs.

major comments (2)
  1. [§4] §4 (Convergence theorem): The argument that constrained Schrödinger solutions converge to the unconstrained Kyle bridge while preserving the equilibrium fixed-point (martingale property for the price process and optimality for the informed trader) requires tightness of the family of measures and passage of the limit inside the optimality condition. The manuscript assumes only continuity or convexity of the cost functions; without uniform integrability or explicit moment bounds, the limit may fail to remain a martingale or satisfy the Kyle optimality, exactly as flagged in the stress-test note.
  2. [§3.1] §3.1 (Representation as Schrödinger problem): The claim that generalized equilibria with default risk are solutions to the constrained bridge problem is central, yet the verification that the default-risk constraint is compatible with the entropy-minimization formulation is only sketched. Explicit construction of the dual potentials or the Radon-Nikodym derivative under the default constraint is needed to confirm the representation is not merely formal.
minor comments (2)
  1. [Introduction] The notation distinguishing the constrained versus unconstrained Schrödinger problems is introduced late; a short table or diagram in the introduction would clarify the hierarchy of problems.
  2. Several standard references on Schrödinger bridges (e.g., the works of Léonard and Nutz) are cited, but the precise relation to the constrained setting could be stated more explicitly to avoid overlap with existing literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and describe the revisions we will make to strengthen the arguments.

read point-by-point responses

  1. Referee: [§4] §4 (Convergence theorem): The argument that constrained Schrödinger solutions converge to the unconstrained Kyle bridge while preserving the equilibrium fixed-point (martingale property for the price process and optimality for the informed trader) requires tightness of the family of measures and passage of the limit inside the optimality condition. The manuscript assumes only continuity or convexity of the cost functions; without uniform integrability or explicit moment bounds, the limit may fail to remain a martingale or satisfy the Kyle optimality, exactly as flagged in the stress-test note.

    Authors: We acknowledge the referee's point that additional justification is needed for the passage to the limit in the convergence theorem. Under the continuity and convexity assumptions on the cost functions, the entropy-minimization property already yields a uniform bound that implies tightness of the family of measures. We will add a new lemma establishing uniform integrability via moment controls derived from convexity of the costs. This will permit interchanging the limit with the optimality conditions, preserving both the martingale property of the price process and the optimality for the informed trader. The revised manuscript will include these explicit bounds and the lemma. revision: yes

  2. Referee: [§3.1] §3.1 (Representation as Schrödinger problem): The claim that generalized equilibria with default risk are solutions to the constrained bridge problem is central, yet the verification that the default-risk constraint is compatible with the entropy-minimization formulation is only sketched. Explicit construction of the dual potentials or the Radon-Nikodym derivative under the default constraint is needed to confirm the representation is not merely formal.

    Authors: We agree that the verification in Section 3.1 can be made fully explicit. We will expand the argument to construct the dual potentials explicitly using the Lagrange multipliers for the default-risk constraint. We will also derive the Radon-Nikodym derivative as the normalized exponential of the potential adjusted by the default indicator function. This will be stated as a detailed proposition showing that the equilibrium measure solves the entropy minimization problem under the constraint, confirming the representation is rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: independent representation and convergence proofs

full rationale

The derivation begins from the externally known Kyle-Brownian bridge link, introduces a generalized class of constrained bridges, proves they solve Schrödinger-type problems via the paper's own arguments, and then establishes convergence of trading-cost equilibria to the classical Kyle model as a consequence of that representation plus limit arguments. No step reduces by definition, fitted parameter, or self-citation chain to its own inputs; the central claims rest on separate mathematical constructions that are not tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete and based on standard background in the area. No explicit free parameters, invented entities, or ad-hoc axioms are mentioned in the provided text.

axioms (1)
  • domain assumption Existence and uniqueness of equilibria in the generalized models with trading costs and default risk
    Implicitly required to state that the bridges correspond to equilibria and that convergence holds.

pith-pipeline@v0.9.0 · 5354 in / 1242 out tokens · 27089 ms · 2026-05-08T06:26:28.945566+00:00 · methodology

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