  In this paper we present a variational calculus approach to Principal-Agent
problem with a lump-sum payment on finite horizon in degenerate stochastic
systems, such as filtered partially observed linear systems. Our work extends
the existing methodologies in the Principal-Agent literature using dynamic
programming and BSDE representation of the contracts in the non-degenerate
controlled stochastic systems. We first solve the Principal's problem in an
enlarged set of contracts defined by a forward-backward SDE system given by the
first order condition of the Agent's problem using variational calculus. Then
we use the sufficient condition of the Agent's problem to verify that the
optimal contract that we obtain by solving the Principal's problem is indeed
implementable (i.e. belonging to the admissible contract set). Importantly we
consider the control problem in a weak formulation. Finally, we give explicit
solution of the Principal-Agent problem in partially observed linear systems
and extend our results to some mean field interacting Agents case.
