Strassen's theorem for biased convex order

2025-09-16

AI Summary

This paper addresses the inadequacy of the classical convex order in characterizing arbitrage-free pricing of American options. To this end, it introduces a “biased convex order” and establishes an equivalent condition for the existence of a biased martingale. Methodologically, the authors employ an integral representation involving compensated Poisson processes and integrate tools from probability theory and convex analysis to extend the classical Strassen theorem to this strengthened stochastic order. The key contribution is twofold: first, it provides the first rigorous characterization of market-consistent pricing measures via the biased convex order; second, it establishes a necessary and sufficient condition linking such measures to the existence of biased martingales. This framework delivers a more refined and economically interpretable theoretical foundation for robust pricing—particularly of higher-order and path-dependent American options—within model-uncertain financial markets.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Auctions and Market-Based Systems

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Extending Strassen's theorem for biased convex order
• Characterizing martingale existence for American option pricing
• Linking biased martingales to strengthened convex order

Innovation

• Extends Strassen's theorem for biased convex order
• Links martingales to strengthened convex order
• Characterizes order via compensated Poisson integrals