Paper: arXiv 2609.18770

Authors: Shuaijie Qian, Guan Qiao

Abstract

This paper studies the regularity of the value function arising from a multidimensional continuous-time principal-agent model with separable, nonquadratic effort costs. The associated stochastic control problem has the output and the agent’s continuation utility as state variables, and its Hamilton-Jacobi-Bellman equation is fully nonlinear and degenerate, with potentially unbounded coefficients. We address these difficulties by adding an independent regularization noise and bounding the effort. For the resulting problem, we establish classical regularity of the value function and show that the optimal effort is unique, positive and remains in a fixed compact subset, uniformly with respect to both the control restriction and the regularization parameter. These estimates allow us first to remove the control restriction and then to let the additional noise vanish. Consequently, we prove that the regularized value function converges to the original value function and conclude that the latter belongs locally to the Sobolev space $W^{2,1}_{\infty, \mathrm{loc}}$, thereby extending the regularity analysis to separable nonquadratic effort costs, for which the arguments yielding classical solutions in the quadratic-cost setting no longer apply.

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 advanced mathematical proofs for a specific type of principal-agent problem. It lacks any empirical validation or data, placing it firmly in the ‘Lab Rats’ quadrant. The novelty comes from extending regularity analysis to nonquadratic effort costs, which is a significant mathematical contribution.

Research Flowchart

  flowchart TD
    A[Research Goal: Regularity of Value Function in Multidimensional P-A Problem] --> B{Methodology: Regularization and Bounding Effort};
    B --> C{Inputs: Multidimensional Continuous-Time P-A Model, Separable Nonquadratic Effort Costs};
    C --> D[Computational Process: Solve HJB Equation with Regularization Noise and Bounded Effort];
    D --> E[Key Findings: Classical Regularity of Regularized Value Function];
    E --> F[Key Findings: Unique, Positive, Bounded Optimal Effort];
    F --> G[Outcomes: Regularized Value Function Converges to Original; Original Value Function is in Sobolev Space $W^{2,1}_{\infty, \mathrm{loc}}$];