arxiv: 2605.11717 · v1 · submitted 2026-05-12 · 💱 q-fin.MF

Recognition: 2 theorem links

· Lean Theorem

On convergence of the Mayer problems arising in the theory of financial markets with transaction cost

Artur Sidorenko, Yuri Kabanov

Pith reviewed 2026-05-13 04:36 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords transaction costsMayer problemoptimal controlconvergencefinancial marketsgeometric formalismportfolio optimizationsolvency cone

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The pith

Optimal values and controls in Mayer problems for transaction-cost markets remain continuous under price approximations.

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

The paper studies the stochastic Mayer control problem of maximizing expected utility of terminal portfolio wealth inside a general geometric framework for markets with proportional transaction costs. Specific models are embedded using a d-dimensional price process S and a cone-valued solvency process K. It proves that both the optimal value and the optimal controls vary continuously when the price process is approximated in a multi-asset setting. A sympathetic reader cares because this robustness means small errors in price data or numerical models do not produce large shifts in best trading strategies.

Core claim

In the geometric formalism defined by the price evolution process S and the cone-valued solvency set process K, the optimal value of the expected-utility maximization problem and the corresponding optimal controls are continuous with respect to approximations of the price process S.

What carries the argument

The geometric formalism that embeds a concrete market model via the d-dimensional price process S and the cone-valued process K describing the evolution of the solvency set, allowing set-valued techniques for optimization.

If this is right

Small changes to observed asset prices produce only small changes in the highest attainable expected utility.
Optimal trading strategies under transaction costs remain close when the price model is slightly perturbed.
Reliable numerical schemes can be used to solve these control problems by working with approximate price processes.
No-arbitrage and hedging criteria derived in the framework inherit the same continuity property.

Where Pith is reading between the lines

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

The continuity property may support the design of stable numerical algorithms for portfolio choice in markets with frictions.
Similar arguments could apply to other stochastic control problems that admit a geometric embedding.
Explicit binomial or trinomial market examples could be used to measure the speed of convergence numerically.

Load-bearing premise

Any given market model of stocks or currencies can be represented inside the abstract framework by suitable choices of the price process S and the solvency cone process K.

What would settle it

A sequence of price processes S_n converging to S such that the maximal expected terminal utility for S_n fails to converge to the maximal expected utility for S would disprove the continuity result.

read the original abstract

The geometric approach to financial markets with proportional transaction cost prescribes to imbed a specific model (of stock market, of currency market etc.), usually given in a parametric form, into a natural framework defined by the two random processes, S and K. The first one, d-dimensional, models the price evolution of basic securities while the second one, cone-valued, describes the evolution of the solvency set. It happened that the fundamental questions -- no-arbitrage criteria, hedging problems, portfolio optimization -- can be studied in this general setting opening the door to set-valued techniques. In this note we explore, in such a general framework, the stochastic Mayer control problem, consisting in the maximization of the expected utility of the portfolio terminal wealth. We get results on continuity of the optimal value and the optimal control under price approximations in a general multi-asset framework described by the geometric formalism.

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

This note extends the (S, K) geometric framework to continuity of the Mayer problem value and optimal controls under price approximations, but stays narrow and abstract.

read the letter

The main point is that Kabanov and Sidorenko take the existing geometric setup with price process S and cone-valued solvency process K and prove that the optimal value and the optimal controls for the stochastic Mayer problem remain continuous when S is approximated. The result applies in a general multi-asset setting without forcing a specific parametric model for stocks or currencies. That is the actual new piece: a continuity statement for this control problem inside the same formalism they already used for no-arbitrage and hedging questions. The paper does well by keeping the argument inside the set-valued techniques already developed for the framework, so no new free parameters or invented objects appear. Once a concrete market is embedded into (S, K), the continuity carries over without extra work. The embedding step is presented as standard rather than a hidden restriction. The soft spots are limited but real. The note is short on explicit conditions: the type of convergence required on the approximating processes, the regularity needed on the utility function, and the precise topology on the controls are not spelled out in the summary. If the full proofs close these points with standard measurable-selection arguments, the claim holds; if they rely on unstated compactness, the result narrows. No internal contradictions or circular definitions show up, and the citation pattern stays within the authors' prior geometric papers. This is work for people already using the (S, K) approach in transaction-cost models. A reader outside that small circle will need the background papers to follow the notation and the set-valued steps. It is not broad enough to change practice in adjacent fields, but within its scope the claim is specific and the framework is reproducible. I would bring it to a reading group only if the group already works on stochastic control with transaction costs. I would not cite it in my own papers in the next year. Still, the central continuity result is well-posed and grounded in prior formal work, so the paper deserves a serious referee rather than a desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the stochastic Mayer problem of maximizing expected utility of terminal portfolio wealth in the general geometric framework for markets with proportional transaction costs. Markets are embedded via a d-dimensional price process S and a cone-valued solvency process K; the central results establish continuity of the optimal value function and of the optimal control with respect to approximations of S.

Significance. If the continuity statements hold, the work supplies a useful robustness property for portfolio optimization under transaction costs that applies uniformly across models embeddable in the (S, K) setting. The reliance on set-valued techniques and the avoidance of parametric restrictions constitute a genuine technical strength, potentially enabling approximation schemes and stability analysis in multi-asset markets.

minor comments (2)

[Abstract] The abstract states continuity results but does not indicate the mode of convergence (e.g., in probability, almost surely, or in a suitable weak topology on controls) or the precise function space in which the price approximations are taken; adding one sentence would improve readability.
The embedding claim for concrete models (stock, currency, etc.) into the (S, K) framework is asserted as natural; a short paragraph or reference to an earlier work illustrating the embedding for at least one standard market would help readers verify that the general setting is not overly restrictive.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the technical strengths of the set-valued approach, and the recommendation for minor revision. The manuscript establishes continuity of the value function and optimal control for the stochastic Mayer problem under approximations of the price process S in the general geometric (S, K) framework.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes continuity of the optimal value and optimal controls for the stochastic Mayer problem under price process approximations within the established geometric (S, K) framework. No equations, fitted parameters, or predictions are presented that reduce by construction to inputs or self-definitions. The framework itself is referenced as prior work but serves as the ambient setting rather than a load-bearing self-citation for the new continuity results, which rely on standard arguments from stochastic control and set-valued analysis. This is a normal non-circular theoretical extension.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the geometric embedding of market models into processes S and K; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Any specific market model can be imbedded into the general framework defined by the price process S and the cone-valued solvency process K.
Explicitly stated as the prescription of the geometric approach in the abstract.

pith-pipeline@v0.9.0 · 5451 in / 1197 out tokens · 65458 ms · 2026-05-13T04:36:24.377881+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
maximization of the expected utility of the portfolio terminal wealth... continuity of the optimal value and the optimal control under price approximations in a general multi-asset framework described by the geometric formalism
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
solvency cone K... cone-valued process... consistent price system Z

Reference graph

Works this paper leans on

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