Viscosity Supersolution Barriers to a Non-local Free Boundary Problem
Abstract
We study a parabolic obstacle partial integro-differential equation (PIDE) with a dynamically moving bilateral free boundary. This type of problem arises in the mathematical modeling of speculative asset bubbles with Lévy 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 Lγ. First, when r+ρ> √λLγ, we prove the global existence of non-negative viscosity supersolutions. Second, in the deficit regime (r+ρ< √λLγ), 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= √λLγ), 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.
Related
- Viscosity and variational approaches in free boundary problems: A Volume in Honor of Sandro Salsa
- Boundary Layers and Initial-Boundary Corner Asymptotics for the 2D Navier-Stokes Equations with Vanishing Vertical Viscosity
- Optimal Reinsurance-Dividend Strategy with Fixed Transaction Costs in a Regime-Switching Brownian Risk Model: A Viscosity Solution to the Impulse Control Problem
- Geometric analysis of a mixed elliptic-parabolic conformally invariant boundary value problem
- The Dirichlet problem with boundary data of fractional smoothness in domains with Ahlfors regular boundary
BibTeX
@misc{arakelyan2026viscosity,
title = {{Viscosity Supersolution Barriers to a Non-local Free Boundary Problem}},
author = {Avetik Arakelyan and Lusine Poghosyan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02381},
eprint = {2609.02381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02381},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Viscosity Supersolution Barriers to a Non-local Free Boundary Problem
AU - Avetik Arakelyan
AU - Lusine Poghosyan
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.02381
UR - https://arxiv.org/abs/2609.02381
AB - We study a parabolic obstacle partial integro-differential equation (PIDE) with a dynamically moving bilateral free boundary. This type of problem arises in the mathematical modeling of speculative asset bubbles with Lévy 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> √λL_γ, we prove the global existence of non-negative viscosity supersolutions. Second, in the deficit regime (r+ρ< √λL_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 C_max 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= √λL_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.
ER -