Paper: arXiv 2609.36405

Authors: Junkee Jeon, Takwon Kim, Jinwan Park, A. Max Reppen

Abstract

We study a finite-horizon reversible investment problem in which a risk-neutral firm adjusts capacity at a proportional purchase cost and a lower salvage value under multi-factor geometric Brownian motion. Via the singular control–optimal switching correspondence, the marginal value of capacity solves a family of parabolic double-obstacle problems. We prove existence, uniqueness and local Sobolev regularity of the strong solution, characterize investment, waiting and disinvestment regions by continuous, strictly separated free boundaries, and verify optimality of the reflected capacity process. Numerically, joint demand improvements shift both boundaries super-additively, 1.5–2.7 times as strongly at the disinvestment boundary, depending on factor correlation.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper features highly advanced mathematics, including singular control, optimal switching, and parabolic double-obstacle problems. While it presents numerical illustrations, it lacks extensive empirical backtesting or real-world data validation, focusing more on theoretical existence and characterization. The multi-factor, finite-horizon reversible investment problem with its detailed mathematical treatment is quite novel.

Research Flowchart

  flowchart TD
    A[Research Goal: Reversible Investment under Multi-Factor Dynamics] --> B{Methodology: Singular Control & Optimal Switching};
    B --> C{Problem Formulation: Parabolic Double-Obstacle Problems};
    C --> D{Inputs: Multi-Factor Geometric Brownian Motion, Proportional Purchase Cost, Salvage Value};
    D --> E{Computational Processes: Existence, Uniqueness, Regularity Proofs, Numerical Simulations};
    E --> F{Key Outcomes: Investment, Waiting, Disinvestment Regions Characterized by Free Boundaries};
    F --> G{Key Finding: Joint Demand Improvements Shift Boundaries Super-Additively (1.5-2.7x stronger at disinvestment)};