  We study a generic principal-agent problem in continuous time on a finite
time horizon. We introduce a framework in which the agent is allowed to employ
measure-valued controls and characterise the continuation utility as a solution
to a specific form of a backward stochastic differential equation driven by a
martingale measure. We leverage this characterisation to prove that, under
appropriate conditions, an optimal solution to the principal's problem exists,
even when constraints on the contract are imposed. In doing so, we employ
compactification techniques and, as a result, circumvent the typical challenge
of showing well-posedness for a degenerate partial differential equation with
potential boundary conditions, where regularity problems often arise.
