Viscosity and Principal-Agent Problem

We develop a stochastic control system from a continuous-time Principal-Agent model in which both the principal and the agent have imperfect information and different beliefs about the project. We consider the agent's problem in this stochastic control system, i.e., we attempt to optimize the agent's utility function under the agent's belief. Via the corresponding Hamilton-Jacobi-Bellman equation the value function is shown to be jointly continuous and to satisfy the Dynamic Programming Principle. These properties directly lead to the conclusion that the value function is a viscosity solution of the HJB equation. Uniqueness is then also established.