Resumen
We consider a problem of finding optimal contracs in continuous time, when the agent's actions are unobservable by 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 contracis, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contrac 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 action 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 suggesta way to check sufficiency using non-convex optimization.