Optimal Contract Design with Quadratic Effort Cost

The existence of an optimal contract of the principal-agent problem is a central issue in contract design. According to Cvitanić et al. [2], such an optimal contract can be derived from the existence of a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which is a degenerate, fully nonlinear parabolic equation. In this work, we follow their model, consider the case with drift control, and prove the existence of the classical solution to the HJB equation.