具有可分离努力成本的多维委托-代理问题的正则性
Regularity of a Multidimensional Principal-Agent Problem with Separable Effort Costs
- The Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究多维连续时间委托-代理模型的值函数正则性，通过添加正则化噪声和限制努力，证明了经典正则性及最优努力唯一性，并扩展到非二次努力成本情形。
AI中文摘要：
本文研究了具有可分离、非二次努力成本的多维连续时间委托-代理模型中所产生的值函数的正则性。相关的随机控制问题以产出和代理人的延续效用作为状态变量，其Hamilton-Jacobi-Bellman方程是完全非线性和退化的，且系数可能无界。我们通过添加独立的正则化噪声并限制努力水平来应对这些困难。对于由此产生的问题，我们建立了值函数的经典正则性，并证明了最优努力是唯一的、正的，且一致地保持在某个固定紧子集中，该一致性同时相对于控制限制和正则化参数成立。这些估计使我们首先能够移除控制限制，然后让附加噪声消失。因此，我们证明了正则化后的值函数收敛于原始值函数，并得出结论：后者局部属于Sobolev空间$W^{2,1}_{\infty, \mathrm{loc}}$，从而将正则性分析扩展到可分离的非二次努力成本，而在二次成本情形下产生经典解的论证不再适用。
英文摘要：
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.
