Paper: arXiv 2307.01986

Authors: Qian Lei, Chi Seng Pun

Abstract

This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium strategies and the associated value functions for time-inconsistent stochastic control problems. Specifically, we consider nonlocality in both time and space, which allows for modelling of the stochastic control problems with initial-time-and-state dependent objective functionals. We leverage the method of continuity to show the global well-posedness within our proposed Banach space with our established Schauder prior estimate for the linearized nonlocal PDE. Then, we adopt a linearization method and Banach’s fixed point arguments to show the local well-posedness of the nonlocal fully nonlinear case, while the global well-posedness is attainable provided that a very sharp a-priori estimate is available. On top of the well-posedness results, we also provide a probabilistic representation of the solutions to the nonlocal fully nonlinear PDEs and an estimate on the difference between the value functions of sophisticated and naïve controllers. Finally, we give a financial example of time inconsistency that is proven to be globally solvable.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical exploration of non-local HJB equations for time-inconsistent stochastic control using sophisticated analytic techniques like Schauder estimates and Banach fixed-point arguments. While mathematically dense and novel in its continuous-time framework, it lacks empirical validation, backtesting, or data analysis beyond a stylized theoretical financial example.

Research Flowchart

  flowchart TD
    A[Goal: Well-posedness of Nonlocal HJB PDEs in Time-Inconsistent Control] --> B[Input: Nonlocal Parabolic PDEs with Time/Space Dependence]
    B --> C[Establish Schauder A-Priori Estimates for Linearized Nonlocal PDE]
    C --> D[Apply Method of Continuity for Global Well-Posedness of Linear Case]
    D --> E[Use Linearization & Banach Fixed-Point Arguments for Nonlinear Case]
    E --> F[Outcomes: Local/Global Solvability, Probabilistic Representation & Sophisticated vs. Naïve Controller Gap]