By Jakša Cvitanic, Jianfeng Zhang

In recent times there was an important bring up of curiosity in continuous-time Principal-Agent versions, or agreement thought, and their functions. Continuous-time versions supply a strong and chic framework for fixing stochastic optimization difficulties of discovering the optimum contracts among events, lower than a number of assumptions at the details they've got entry to, and the impression they've got at the underlying "profit/loss" values. This monograph surveys contemporary result of the idea in a scientific manner, utilizing the technique of the so-called Stochastic greatest precept, in types pushed through Brownian movement. optimum contracts are characterised through a approach of Forward-Backward Stochastic Differential Equations. In a few fascinating distinctive situations those will be solved explicitly, allowing derivation of many qualitative financial conclusions.

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**Extra info for Contract Theory in Continuous-Time Models**

**Example text**

The agent’s problem is, given (CT , c), VA (CT , c) := sup VA (CT , c, u) u∈U T := sup E u u∈U = sup E MTu u∈U uA t, (X)t , ct , ut dt + UA (X)T , CT 0 T uA t, (X)t , ct , ut dt + UA (X)T , CT . 5) 0 Here, (X)t denotes the path of X up to time t, U is the admissible set of the agent’s controls u, which are F-adapted processes taking values in some set U ⊂ Rk for some k. We will specify the technical requirements on U later. We note that X is a fixed process which does not change its values with the choice of u.

1 Linear Dynamics and Control of Volatility 23 That is, if the principal and agent have the same power utility, and they both behave optimally, the payoff turns out to be linear at time T . 3, hence the linear contract implements the first best. 1 (Nonlinear Payoff as the ODE Solution) Assume that UA (x) = log(x) and UP (x) = −e−x . Recall the special function Ei(x), called exponential integral, defined by Ei(x) := − ∞ −x e−t dt. 18) This is a well defined function for x < 0, and it is continuous and decreases from 0 to −∞.

The economic theory of agency: the principal’s problem. Am. Econ. Rev. 63, 134–139 (1973). Papers and Proceedings of the Eighty-fifth Annual Meeting of the American Economic Association Chapter 4 The General Risk Sharing Problem Abstract In this chapter we consider general diffusion dynamics for the output process with a general cost function depending on the agent’s actions and/or the output values. The main qualitative conclusions from the case of linear drift dynamics of the previous section still hold true with nonlinear drift dynamics—a linear benchmark contract is optimal, and it implements the first best actions.