Dynamic Principal-Agent Problems

We consider a continuous-time setting, in which the agent can control both the drift and the volatility of the underlying process. The principal can observe the agent’s action and can offer payment at a continuous rate, as well as a bulk payment at the end of the fixed time horizon. In examples, we show that if the principal and the agent have the same CRRA utility, or they both have (possibly different) CARA utilities, the optimal contract is (ex-post) linear; if they have different CRRA utilities, the optimal contract is nonlinear, and can be of the call option type. We use martingale/duality methods, which, in the general case, lead to the optimal contract as a fixed point of a functional that connects the agent’s and the principal’s utility maximization problems.