Viscosity Supersolution Barriers to a Non-local Free Boundary Problem

Battle Count: 80

The paper is highly relevant to the theoretical foundations of speculative bubble modeling in financial markets, particularly in understanding when bubbles can persist versus when they must burst. The Lévy jump-diffusion framework captures realistic market dynamics with discontinuous price movements. However, the paper is purely theoretical and does not provide directly implementable trading strategies, pricing algorithms, or numerical methods. Its relevance is primarily at the level of understanding the mathematical conditions governing bubble dynamics, which could inform risk models and bubble detection frameworks. The critical horizon threshold concept has potential implications for timing bubble exits, but translating the abstract PIDE results into practical trading signals would require significant additional work.

This is a highly theoretical pure mathematics paper requiring deep expertise in viscosity solution theory, non-local integro-differential equations, Lévy processes, and optimal stopping theory. The constructions involve careful piecewise polynomial profiles, Taylor expansions for jump operators, Cauchy-Schwarz and Minkowski inequalities for Lévy integrals, and asymptotic slope envelope arguments. There is no code or numerical implementation provided. Reproducing the results requires advanced mathematical training rather than computational resources.

This is a pure mathematics paper with complete, self-contained proofs. All assumptions, definitions, and theorems are explicitly stated. The constructions are fully specified with explicit formulas for the supersolution barriers. No computational experiments or external data are required. The mathematical arguments can be independently verified from the paper alone.