Optimal Quitting in a Dynamic Agency Problem for Jump Diffusions

Abstract

In this paper, we study the agent's optimal quitting in a continuous-time principal-agent problem, where the agent is payed once at the end of the contract. In a jump diffusion setting, we formulate the agency problem as a combined optimal stopping and stochastic control problem in weak formulation. To find the solutions, we develop the classical verification theorem in terms of Variational Inequality Hamilton-Jacobi-Bellman (VIHJB) equations in weak formulation. Finally, we solve explicitly the VIHJB equations in a special case.