AI 中文总结
该研究针对带 Lévy 跳过程的投机资产泡沫建模对应的非局部自由边界问题，分析不同参数区间下粘性上解的存在性，推导临界时间阈值等关键结论，明确上解障碍的构建条件。
AI 中文摘要
我们研究一类具有动态移动双边自由边界的抛物型障碍偏积分-微分方程（PIDE），这类问题源于带 Lévy 跳过程的投机资产泡沫的数学建模。我们考虑在三类不同参数区间内，具有线性渐近增长（无穷远处为 $O(|g|)$）的函数类中粘性上解的存在性，旨在通过分析由贴现率 $r$ 和均值回复率 $\rho$ 定义的稳定局部漂移，以及由大跳强度 $\text{lambda}$ 和 Lipschitz 常数 $L_\gamma$ 表征的非局部跳扩散之间的平衡，确定何时可构建此类上解障碍。首先，当 $r+\rho > \sqrt{\lambda}L_\gamma$ 时，我们证明非负粘性上解的全局存在性；其次，在赤字区间（$r+\rho < \sqrt{\lambda}L_\gamma$）中，我们证明有限时间区间上的存在性并推导临界时间阈值 $T_{\text{crit}}$，利用渐近斜率包络证明，在 $T_{\text{crit}}$ 之外不存在非负线性增长上解；最后，在精确临界边界（$r+\rho = \sqrt{\lambda}L_\gamma$）处，只要满足额外的空间无交叉条件，我们通过构造光滑上解证明其全局存在性。
英文摘要
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 $r+ρ> \sqrtλL_γ$, we prove the global existence of non-negative viscosity supersolutions. Second, in the deficit regime ($r+ρ< \sqrtλL_γ$), 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 ($r+ρ= \sqrtλL_γ$), 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.
Comments28 pages; Keywords: Viscosity solutions, Financial bubbles, Lévy processes, Free boundary
