奈特不确定下考虑逆向选择的最优动态契约设计

现实世界中不确定可以分成概率不确定和奈特（Knight）不确定.本文从奈特不确定框架研究连续时间下面临道德风险的委托-代理问题，试图分析低道德水平代理人逆向选择对高道德水平代理人延续价值和动态最优契约执行过程的影响.首先，基于非线性期望理论，利用G-布朗运动刻画契约项目盈利的动力学方程，进而定义了委托人和代理人的期望效用.其次，用带G-布朗运动驱动的随机微方程刻画了高道德水平代理人和低道德水平代理人价值过程.并且，利用次线性期望下的最优性原理，推导出高道德水平代理人预期效用值函数所满足的哈密尔顿-雅可比-贝尔曼（HJB）方程，在低道德水平代理人逆向选择下给出高道德水平代理人获得的最优报酬，同时得到低道德水平与高道德水平两类代理人最优努力水平的隐式表达式.最后，通过Matlab数值模拟，阐释了奈特不确定对委托人和不同类型代理人的策略影响.模拟分析结果表明，低道德水平代理人的逆向选择将会直接影响高道德水平代理人的契约支付，进而会影响契约有效性；而作为造成低道德水平代理人逆向选择的重要原因，奈特不确定也使得委托代理双方的决策更加谨慎.

Contract design with Knightian uncertainty and adverse selection

This paper studies a principal-agent problem in continuous time with ambiguity based on that the uncertainty of probabilistic model is nature. We put an emphasis on the effects of the bad moral agent's adverse selection on the contract and the continuation value of the good moral agent. Firstly, we establish the dynamic equations of the bad moral agent's continuation value utility as well as the good moral agent's total utility. Then, by the theory and methods of stochastic optimal control under the premise of sublinear expectation, we deduce the corresponding Hamilton-Jacobi-Bellman (HJB) equation of the good moral agent's value function, and the implicit expression of the optimal principal's payment. Finally, the numerical simulations are provided for explaining the effects of Knightian uncertainty on the optimal dynamic contract. The results show that, the bad moral agent's adverse selection would reduce the optimal payment of the good moral agent directly, and then it would influence the availability of contract. Moreover, as the important reason of the adverse selection, the Knightian uncertainty would make the principal-agent decisions more cautious.