Mathematical Theory for General Moral Hazard Problems

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This chapter describes a general theory of optimal contracting with hidden or non-contractable actions in continuous-time, developed by applying the stochastic maximum principle. The main modeling difference with respect to the full information case is that we will now assume that the agent controls the distribution of the output process with his effort. Mathematically, this is modeled using the so-called “weak formulation” and “weak solutions” of the underlying SDEs. Necessary and sufficient conditions are derived in terms of the so-called adjoint processes and corresponding Forward-Backward SDEs. These processes typically include the output process, the agent’s expected utility process, the principal’s expected utility process, and the ratio of marginal utilities process.