Viscosity Supersolution Barriers to a Non-local Free Boundary Problem

We study a parabolic obstacle partial integro-differential equation with a dynamically moving bilateral free boundary. This type of problem arises in the mathematical modeling of speculative asset bubbles with L\'evy jump processes. We investigate the existence of viscosity supersolution barriers within the class of functions exhibiting linear asymptotic growth (O(g) at infinity) across three distinct parametric regimes. Our intention is to determine when such a barrier can be constructed by analyzing the balance between the stabilizing local drift, defined by the discount rate r and mean-reversion ρ, and the non-local jump dispersion, characterized by the large-jump intensity λ and Lipschitz constant L_γ. First, when r+ρ > √_γ, we prove the global existence of non-negative viscosity supersolutions. Second, in the deficit regime (r+ρ < √_γ), we construct non-negative supersolutions for every finite horizon $T>0$. However, by utilizing an asymptotic slope envelope, we prove that these barriers cannot be bounded by a fixed, pre-determined linear growth ceiling Cₘₐₓ across arbitrarily large horizons; rather, the required linear growth constant must inflate exponentially as the horizon length increases. Finally, at the exact critical boundary (r+ρ = √_γ), we show the existence of a supersolution with a uniform spatial growth bound, provided an additional spatial no-crossing condition holds on the negative tail.