This chapter reviews four principal-agent models in continuous time. The first model is about a contract problems with full information, which is known as a risk-sharing problem. The second model is concerned with optimal contract problems with hidden actions and the payment to the agent is lump-sum at the end of the contract. The third model is similar as the second one, but the payment to the agent is continuous. The last model is about a concrete problem–the optimal insurance design problem, in which both the insurer and the insured are subject to Knightian uncertainty about the loss distribution, and the Knightian uncertainty is modeled in a g-expectation framework. In this problem, the endogenous characterization of the optimal indemnity extends the classical theorems of Arrow and Raviv in the classical situation. In the presence of Knightian uncertainty, the optimal insurance contract is shown to be not only contingent on the realized loss but also on another source of uncertainty coming from the ambiguity.
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