Paper: arXiv 2501.17577

Authors: Giorgio Ferrari, Neofytos Rodosthenous

Abstract

We study a class of singular stochastic control problems for a one-dimensional diffusion $X$ in which the performance criterion to be optimised depends explicitly on the running infimum $I$ (or supremum $S$) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process $(X,I)$ or $(X,S)$; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager’s time-preferences depend on the company’s historical worst performance.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of singular stochastic control with running extrema and introduces novel integral operators to handle multi-dimensional jump dynamics. However, as a purely theoretical manuscript, it completely lacks empirical backtesting, real-world data validation, or code implementations.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve 2D singular control problems depending on running infimum/supremum] --> B[Inputs: 1D diffusion X, running extrema I/S, and state-dependent cost functions]
    B --> C[Methodology: Introduce two novel integral operators consistent with HJB equation]
    C --> D[Verification: Prove a general Verification Theorem for 2D singular stochastic control]
    D --> E[Application: Apply framework to solve optimal dividend problem with historical worst performance]
    E --> F[Outcome: Explicit optimal dividend policy and value function under time-preference constraints]