Some PDE results in Heston model with applications

Paper: arXiv 2504.19859

Authors: Edoardo Lombardo

Abstract

We present here some results for the PDE related to the logHeston model. We present different regularity results and prove a verification theorem that shows that the solution produced via the Feynman-Kac theorem is the unique viscosity solution for a wide choice of initial data (even discontinuous) and source data. In addition, our techniques do not use Feller’s condition at any time. In the end, we prove a convergence theorem to approximate this solution by means of a hybrid (finite differences/tree scheme) approach.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is heavily mathematical, featuring advanced PDE theory, viscosity solutions, and rigorous stochastic analysis without relying on the Feller condition, indicating high math complexity. However, it lacks empirical backtesting, dataset usage, or implementation details for the hybrid scheme, focusing on theoretical convergence rather than practical trading readiness.

Research Flowchart

  flowchart TD
  A["Research Goal: Analyze PDE for logHeston model & prove solution uniqueness"] --> B{"Key Methodology"}
  
  B --> C["Feynman-Kac Theorem & Viscosity Solution Analysis"]
  B --> D["Hybrid Computational Scheme<br/>(Finite Differences + Tree)"]
  
  C --> E["Key Inputs/Assumptions"]
  E --> F["Discontinuous Initial & Source Data"]
  E --> G["No Feller Condition Required"]
  
  F & G --> H["Main Findings & Outcomes"]
  D --> H
  
  H --> I["Regularity Results for PDE"]
  H --> J["Uniqueness Proof for Viscosity Solution"]
  H --> K["Convergence Theorem for Approximation Scheme"]