Coordinate-covariant orientation laws for no-trade regions

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

In portfolio control, a utility function quantifies preferences over wealth or holdings, and transaction costs charge changes in positions. A no-trade region is the set of states where immediate trading is suboptimal. Small-cost asymptotics study how this region scales as a cost parameter tends to zero. A parallelotope is an affine image of a box; its orientation is determined by its edge or normal directions.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The contribution reports a conjectured parallelotope shape without fixing asset dynamics, utility, cost scaling, or coordinates. These choices are necessary for a specific asymptotic orientation result. The supplied catalogue marks this record partially solved; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.

Statement recorded in the supplied source.

A.10 Ostrov, Daniel

My primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected "utility" of the stocks is maximized. (The term "utility" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a "hold region" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be. [S2]

Source status.

The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

The source asks about the shape and orientation of a small-cost no-trade region.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001175 (AIM-COMPUTATION-0013). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AIM Workshop Problem Lists, source linked by catalogue record AIM-COMPUTATION-0013. https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20001175 and code AIM-COMPUTATION-0013 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The participant contribution in the primary AIM optimal-control workshop PDF was consulted. Its broad scope is retained above. The following explicitly chosen finite or smooth model tests one route suggested by the agenda; it is not a reconstruction of a missing theorem. The elementary arguments are not claimed novel, and no current-resolution audit of the whole field is asserted.

Research attempt: A static no-trade parallelogram is not aligned by covariance alone

The source's dynamic small-cost claim lacks a fully specified price process, objective and cost model. Choose instead the following explicit static surrogate: for a positive-definite matrix $H$, target $y$ and $\lambda>0$, minimize \[ F(q)=\tfrac12(q-y)^TH(q-y)+\lambda\sum_i|q_i|. \] Starting from zero holdings, the no-trade condition $q=0$ is optimal exactly when $\|Hy\|_\infty\le\lambda$. Indeed $F(q)-F(0)=q^THq/2-q^THy+\lambda\sum|q_i|$, nonnegative under the stated inequalities. If a coordinate of $Hy$ has magnitude larger than $\lambda$, take a sufficiently small step $q$ along that signed coordinate; its negative linear term dominates the positive quadratic term, so zero is not optimal. Strict positive definiteness gives uniqueness of the minimizer.

Thus in this surrogate the no-trade set is $H^{-1}[-\lambda,\lambda]^d$. In two dimensions it is a parallelogram with edge directions $H^{-1}e_1,H^{-1}e_2$. These generally differ from the eigendirections of $H$. For $H=\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)$, its edge directions are $(2,-1)$ and $(-1,2)$, whose dot product is $-4$, while the eigendirections $(1,1),(1,-1)$ are orthogonal. Hence choosing covariance eigenvectors by inspection does not even solve this simpler orientation problem.

Under an invertible coordinate change $q'=Aq$, the quadratic matrix changes to $A^{-T}HA^{-1}$, but the transaction penalty changes to $\lambda\|A^{-1}q'\|_1$. Retaining the original axis-aligned $\ell^1$ penalty after rotating would define another optimization problem. Substituting the coordinates in $F$ proves these transformation rules directly.

This is a diagnostic of how curvature and the cost geometry jointly determine orientation. It neither proves that a stochastic-control no-trade region is asymptotically a parallelogram nor identifies its effective curvature matrix.

Verification and adversarial audit

Necessity and sufficiency of the static no-trade inequalities were proved. The inverse matrix and nonorthogonal edge directions were checked. The limit $\lambda=0$ contracts this surrogate region to zero, consistent with strict convexity, but no stochastic scaling law is inferred.

Audit corrections.

The static surrogate was labelled explicitly; its parallelogram was not used as a proof of the source dynamic asymptotic claim.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The first gap is a specified dynamic model and a proved small-transaction-cost limit identifying the relevant region and tensors. The entire asymptotic shape and orientation question in the source remains separate from the static example.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.