The principal-agent problem is a classic problem in economics, in which the principal seeks an optimal way to delegate a task to an agent that has private information or hidden action. A general continuous-time stochastic control problem based on the moral hazard problem in Sannikov (2008) is considered, with more general retirement cost and structure. In the problem, a risk-neutral principal tries to determine an optimal contract to compensate a risk-averse agent for exerting costly and hidden effort over an infinite time horizon. The compensation is based on observable output, which has a drift component equal to the hidden effort and a noise component driven by a Brownian motion.

In this thesis, a rigorous mathematical formulation is posed for the problem, which is modeled as a combined optimal stopping and control problem. Conditions are given on how a solution to the control problem could be implemented as a contract in the principal-agent framework with moral hazard. Our formulation allows for general continuous retirement profit functions, subject to an upper bound by the first-best profit. The optimal profit function is studied and proved to be concave and continuous. It is shown that the optimal profit function is the unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation.