Superreplication in Large-Trader Market Models

Definitions and mathematical statement

In the large-trader model of the report, trading strategies affect asset prices. Under its assumptions excluding arbitrage for bounded semimartingale strategies, determine which contingent claims can be replicated or superreplicated, and give the corresponding necessary-and-sufficient conditions.

To describe the target without inventing the model, let $V_T(x,\vartheta)$ denote its terminal wealth for initial capital $x$ and an admissible strategy $\vartheta$. For a cash claim $H$ at the model's terminal time, the usual replication and superreplication conditions would be, respectively, \[ \exists\vartheta:\quad V_T(x,\vartheta)=H\quad\text{almost surely}, \] \[ \exists\vartheta:\quad V_T(x,\vartheta)\ge H\quad\text{almost surely}. \] These formulas only explain the desired attainable-claim characterization. The source does not specify whether capital $x$ is fixed, optimized, or merely required to be finite, nor whether settlement is cash, physical delivery, or liquidation at an impact-dependent value.

The filtered probability space, price-impact functional, self-financing rule, admissibility and wealth lower bounds, horizon, claim integrability, and the exact no-arbitrage conditions are missing. The source's bounded-semimartingale condition is retained for the arbitrage setup; it does not by itself define the entire hedging class. No linear wealth formula for an unaffected price process or small-trader martingale-measure criterion is silently substituted.

The task includes exact replication and dominating superreplication, which are distinct. It does not assert that every claim is hedgeable or prescribe a dual formula. The supplied verification identifies the request as reconstructed and unverified as a standalone characterization theorem, even though the report confirms open hedging questions in its specific setup. The draft therefore records the missing financial-model definitions rather than claiming a universal solution for all price-impact markets.

Short English statement

The source asks for attainable-claim and superhedgeable-claim characterizations in its price-impact model under no-arbitrage assumptions for bounded semimartingale trading. The model, admissibility, settlement, and initial-capital conventions are missing; the request is explicitly reconstructed rather than a verified standalone theorem.

Sources

• Local UnsolvedMath registry, permanent ID 30000915, code catalogue entry, "Superreplication in Large-Trader Market Models." Proposed year 2008; proposer unspecified.
• Stochastic Analysis in Finance and Insurance (2008), Oberwolfach Reports, pp. 173--240, DOI 10.4171/owr/2008/04; https://doi.org/10.4171/owr/2008/04.
• Supplied "Super-replication in large financial markets": https://doi.org/10.1016/j.spa.2005.06.010. Its precise relation to this large-trader price-impact setup is not supplied.
• Supplied "Bank--Baum": https://doi.org/10.18452/3499.
• Statement status reconstructed_unverified. The supplied verification says the report confirms open replication/superreplication questions under its assumptions but does not give the extracted request as a precise standalone characterization theorem.
• Literature triage checked 2026-08-21: partially solved. It reports dual characterizations and existence results in important semimartingale and discrete-time models, without a complete solution under the report's assumptions. Exact model comparisons and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.