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

Randomisation with moral hazard: a path to existence of optimal contracts