Principal-Agent Problems in Continuous Time
Motivated by the problems of optimal compensation of executives and of investment fund managers, we consider principal-agent problems in continuous time, when the principal's and the agent's risk-aversion are modeled by standard utility functions. The agent can control both the drift (the ``mean") and the volatility (the ``variance") of the underlying stochastic process. The principal decides what type of contract/payoff to give to the agent. We use martingale/duality methods familiar from the theory of continuous-time optimal portfolio selection. Our results depend on whether the agent can control the drift independently of the volatility, or not, and whether they have the same utility functions. We get the following results in illustrative examples of our general theory: if both the agent and the principal have the same power utility, or they both have (possibly different) exponential utilities, the optimal contract is (ex-post) linear; if they have different power utilities, the optimal contract is nonlinear. We also present an example in which a call option-type contract is optimal. Finally, we establish an approach for solving the principal-agent problem in very general models and with a general cost function, and show how the approach works in non-trivial examples.