Caratheodory, Finite Resources and the Geometry of Arbitrage

TL;DR

This paper explores the geometric parallels between thermodynamics and finance, showing that resource limitations and topological constraints lead to the exponential family as a unifying structure for understanding arbitrage and state accessibility.

Contribution

It establishes a novel geometric framework linking thermodynamic adiabatic inaccessibility with financial arbitrage constraints, highlighting the exponential family as fundamental in both fields.

Findings

Exponential family characterizes resource-limited systems in thermodynamics and finance.

Topological restrictions prevent certain state transitions, analogous in both domains.

Resource constraints underpin the geometric structure of arbitrage and thermodynamic states.

Abstract

Caratheodory's axiom of adiabatic inaccessibility states that, in any neighborhood of a thermodynamic state, certain states remain unreachable via adiabatic processes. Non-arbitrage mirrors this topological restriction in finance. Preserving this constraint in resource-limited systems identifies the exponential family not as a modeling convenience but as the requisite geometric structure unifying both domains.