Strassen's theorem for biased convex order

Paper: arXiv 2509.13041

Authors: Beatrice Acciaio, Mathias Beiglböck, Evgeny Kolosov, Gudmund Pammer

Abstract

Strassen’s theorem asserts that for given marginal probabilities $μ,ν$ there exists a martingale starting in $μ$ and terminating in $ν$ if and only if $μ,ν$ are in convex order. From a financial perspective, it guarantees the existence of market-consistent martingale pricing measures for arbitrage-free prices of European call options and thus plays a fundamental role in robust finance. Arbitrage-free prices of American options demand a stronger version of martingales which are ‘biased’ in a specific sense. In this paper, we derive an extension of Strassen’s theorem that links them to an appropriate strengthening of the convex order. Moreover, we provide a characterization of this order through integrals with respect to compensated Poisson processes.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper provides a rigorous mathematical extension of Strassen’s theorem to biased convex orders, addressing American option pricing in robust finance. While it features advanced probability theory, proofs, and strong theoretical novelty, it is entirely theoretical with no empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Strassen's Theorem<br>for American Option Pricing] --> B[Inputs: Marginal Probabilities μ, ν &<br>Biased Martingale Conditions]
    B --> C[Methodology: Strengthen Convex Order<br>for Biased Martingales]
    C --> D[Analysis: Integrate wrt Compensated<br>Poisson Processes]
    D --> E[Outcome 1: Extended Strassen's Theorem<br>for Biased Convex Order]
    D --> F[Outcome 2: Characterization via<br>Compensated Poisson Integrals]