By Jakša Cvitanic, Jianfeng Zhang
In recent times there was an important bring up of curiosity in continuous-time Principal-Agent types, or agreement concept, and their purposes. Continuous-time types offer a robust and stylish framework for fixing stochastic optimization difficulties of discovering the optimum contracts among events, lower than quite a few assumptions at the info 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 approach, utilizing the strategy of the so-called Stochastic greatest precept, in types pushed by way of Brownian movement. optimum contracts are characterised through a approach of Forward-Backward Stochastic Differential Equations. In a few fascinating distinctive circumstances those could be solved explicitly, permitting derivation of many qualitative monetary conclusions.
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Additional resources for Contract Theory in Continuous-Time Models
Proof Introduce the agent’s objective function, given a contract F (XT ) − DT : Y˜tu,v := Et UA F XTu,v − DT , Gu,v T − = UA F XTu,v − DT , Gu,v T T t Z˜ su,v dBs . 23): ⎧ t ∗=x+ ⎪ ˜ ⎪ b∗ s, X˜ s∗ , Y˜s1,∗ , Y˜s2,∗ , Z˜ s1,∗ ds X ⎪ t ⎪ ⎪ 0 ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ I2 s, X˜ s∗ , Y˜s1,∗ , Y˜s2,∗ , Z˜ s1,∗ dBs ; + ⎪ ⎪ ⎪ 0 ⎪ ⎪ t ⎪ ⎪ ⎪ ˜ ∗t = ⎪ g ∗ s, X˜ s∗ , Y˜s1,∗ , Y˜s2,∗ , Z˜ s1,∗ ds; G ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ T ⎪ ⎪ 1,∗ ⎪ ˜∗ − Z˜ s∗ dBs ; ⎨ Y˜t∗ = λUA X˜ T∗ − IP YT , G T t T ⎪ ⎪ ⎪ ˜ ∗ F X˜ ∗ − ˜t1,∗ = ∂C UA F X˜ ∗ − DT , G ⎪ Y Z˜ s1,∗ dBs ⎪ T T T ⎪ ⎪ t ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ (∂x b)∗ s, X˜ s∗ , Y˜s1,∗ , Y˜s2,∗ , Z˜ s1,∗ Y˜s1,∗ + ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ + (∂x g)∗ s, X˜ s∗ , Y˜s1,∗ , Y˜s2,∗ , Z˜ s1,∗ Y˜s2,∗ ds; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎩ Y˜t2,∗ = ∂G UA F X˜ ∗ − DT , G ˜∗ − Z˜ s2,∗ dBs .
However, we derive general results from scratch in Part V of the book, and apply them here to the problem at hand. 8) (u,v)∈A where Y0u,v = E Uλ XTu,v , Gu,v . T This formulation falls under the framework of Sect. 2, with higher dimensions. 9) t = ⎪ 0 ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ u,v ⎪ ⎩ Yt = Uλ (XTu,v , Gu,v Zsu,v dBs . T )− t The first two equations are in the form of standard SDEs, and the third one is a BSDE: instead of an initial condition, a terminal condition YTu,v = Uλ XTu,v , Gu,v T is imposed, and the solution consists not just of process Y , but of a pair of adapted processes (Y u,v , Z u,v ).
Stud. 16, 173–208 (2003) Part III Second Best: Contracting Under Hidden Action—The Case of Moral Hazard Chapter 5 Mathematical Theory for General Moral Hazard Problems Abstract 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.