Paper: arXiv 2511.18169

Authors: Atiqah Almuzaini, Çağın Ararat, Jin Ma

Abstract

We revisit the well-studied superhedging problem under proportional transaction costs in continuous time using the recently developed tools of set-valued stochastic analysis. By relying on a simple Black-Scholes-type market model for mid-prices and using continuous trading schemes, we define a dynamic family of superhedging sets in continuous time and express them in terms of set-valued integrals. We show that these sets, defined as subsets of Lebesgue spaces at different times, form a dynamic set-valued risk measure with multi-portfolio time-consistency. Finally, we transfer the problem formulation to a path-space setting and introduce approximate versions of superhedging sets that will involve relaxing the superhedging inequality, the superhedging probability, and the solvency requirement for the superhedging strategy with a predetermined error level. In this more technical framework, we are able to relate the approximate superhedging sets at different times by means of a set-valued Bellman’s principle, which we believe will pave the way for a set-valued differential structure that characterizes the superhedging sets.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous theoretical framework using advanced set-valued stochastic analysis to address the superhedging problem under continuous-time transaction costs. While it introduces strong mathematical novelty and dynamic programming principles, it is purely theoretical with no empirical backtesting or data applications.

Research Flowchart

  flowchart TD
    A[Goal: Analyze Continuous-Time Superhedging<br/>with Proportional Transaction Costs] --> B[Market Inputs:<br/>Black-Scholes Mid-Prices & Continuous Trading]
    B --> C[Construct Dynamic Superhedging Sets via<br/>Set-Valued Stochastic Integrals]
    C --> D[Prove Properties: Dynamic Set-Valued Risk Measure<br/>with Multi-Portfolio Time-Consistency]
    D --> E[Path-Space Reformulation:<br/>Introduce Approximate Superhedging Sets with Error Levels]
    E --> F[Outcomes: Establish Set-Valued Bellman's Principle<br/>Paving Way for Differential Characterization]

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