Viscosity Supersolution Barriers to a Non-local Free Boundary Problem

Paper: arXiv 2609.02381

Authors: Avetik Arakelyan, Lusine Poghosyan

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 consider the existence of viscosity supersolutions 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 supersolution barrier can be built 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 prove existence on finite horizons and derive a critical horizon threshold $T_{\mathrm{crit}}$. Utilizing an asymptotic slope envelope, we prove that no non-negative linear-growth supersolution can exist beyond $T_{\mathrm{crit}}$. Finally, at the exact critical boundary ($r+ρ= \sqrtλL_γ$), we show global existence by constructing a smooth supersolution, provided an additional spatial no-crossing condition holds.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on the existence of viscosity supersolutions for a complex PIDE. It demonstrates significant mathematical depth and extends existing models to include Lévy jump processes, showing good novelty. However, it lacks any empirical validation or data analysis, placing it firmly in the theoretical domain.

Research Flowchart

  flowchart TD
    A[Research Goal: Viscosity Supersolution Barriers for PIDE] --> B{Parametric Regimes Analysis};

    B --> C1{Regime 1: r+ρ > √λLγ};
    C1 --> D1[Outcome 1: Global Existence of Non-negative Supersolutions];

    B --> C2{Regime 2: r+ρ < √λLγ (Deficit)};
    C2 --> D2[Outcome 2: Existence on Finite Horizons & Critical Horizon T_crit];
    D2 --> E2[Outcome 2b: No Linear-Growth Supersolution Beyond T_crit];

    B --> C3{Regime 3: r+ρ = √λLγ (Critical Boundary)};
    C3 --> D3[Outcome 3: Global Existence via Smooth Supersolution (if no-crossing holds)];

    style A fill:#DDEBF7,stroke:#336699,stroke-width:2px,color:#000
    style B fill:#FBE4C0,stroke:#B36B00,stroke-width:2px,color:#000
    style C1 fill:#E6F3EA,stroke:#4CAF50,stroke-width:2px,color:#000
    style C2 fill:#FEECEB,stroke:#F44336,stroke-width:2px,color:#000
    style C3 fill:#E6E6FA,stroke:#8A2BE2,stroke-width:2px,color:#000
    style D1 fill:#D4EDDA,stroke:#28A745,stroke-width:2px,color:#000
    style D2 fill:#F8D7DA,stroke:#DC3545,stroke-width:2px,color:#000
    style E2 fill:#F8D7DA,stroke:#DC3545,stroke-width:2px,color:#000
    style D3 fill:#D7D7F8,stroke:#6A5ACD,stroke-width:2px,color:#000

Cited References (39)

1. A non-local free boundary problem arising in a theory of financial bubbles (2014)
2. Overconfidence and speculative bubbles (2003)
3. L 'e vy Processes in Finance: Pricing Financial Derivatives (2003)
4. Asset price bubbles from heterogeneous beliefs about mean reversion rates (2004)
5. Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited (2008)
6. Option valuation using the fast Fourier transform (1998)
7. The fine structure of asset returns (2002)
8. Optimal stopping and perpetual options for L 'e vy processes (2002)

0 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.