Superhedging Supermartingales

TL;DR

This paper extends classical supermartingale theory to a non-probabilistic setting using superhedging operations, providing new decomposition and convergence results without requiring integrability.

Contribution

It introduces a novel framework replacing expectation with subadditive operators motivated by no-arbitrage pricing, extending supermartingale theory beyond probabilistic models.

Findings

Proves Doob's supermartingale decomposition in the new setting

Establishes convergence theorems for nonnegative supermartingales

Shows how to characterize martingales via superhedging properties

Abstract

Supermartingales are here defined on a non-probabilistic setting and can be interpreted solely in terms of superhedging operations. The classical expectation operator is replaced by a pair of subadditive operators one of them providing a class of null sets and the other one acting as an outer integral. These operators are motivated by a financial theory of no-arbitrage pricing. Such a setting extends the classical stochastic framework by replacing the path space of the process by a trajectory set, while also providing a financial/gambling interpretation based on the notion of superhedging. The paper proves analogues of the following classical results: Doob's supermartingale decomposition and Doob's pointwise convergence theorem for nonnegative supermartingales. The approach shows how linearity of the expectation operator can be circumvented and how integrability properties in the proposed setting lead to the special case of (hedging) martingales while no integrability conditions are required for the general supermartingale case.