We consider a problem of finding optimal contracts in continuous time, when the agent's actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal's utility is infinite, while it becomes finite with hidden actions, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization.

We analyze a novel principal-agent problem of moral hazard and adverse selection in continuous time.The constant private shock revealed at time zero when the agent selects the contract has a long-term impact on the optimal contract. The latter is based not only on the continuation value of the agent who truthfully reports, but also contingent upon the continuation value of the agent who misreports, called temptation value. The good agent is retired when the temptation value of the bad agent becomes large, because then it is expensive to motivate the good agent. The bad agent is retired when the temptation value of the good agent becomes small, because then the future payment does not provide sufficient incentives. We also compare the efficiency of the shutdown contract and the screening contract and find that the screening contract can bring more profit to the principal only when the agent's reservation utility is sufficiently small.

We present a unified approach to solving contracting problems with full information in models driven by Brownian motion. We apply the stochastic maximum principle to give necessary and sufficient conditions for contracts that implement the so-called first-best solution. The optimal contract is proportional to the difference between the underlying process controlled by the agent and a stochastic, state-contingent benchmark. Our methodology covers a number of frameworks considered in the existing literature. The main finance applications of this theory are optimal compensation of company executives and of portfolio managers.