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## Optimal contracts in continuous-time models

Venue: | J. Appl. Math. Stoch. Anal. 2006 (2006), Article ID |

Citations: | 5 - 1 self |

### Citations

1234 | Optimization by Vector Space Methods
- Luenberger
- 1969
(Show Context)
Citation Context ...s an implementable contract F such that uF = u, vF = v, and F(ω,Xu,v)= C. Then obviously V2 =V 2. (2.14) To simplify the notations, from now on we abuse the notation and use V2 to denote the right-hand side of (2.12). Also for simplicity, henceforth we use the notation for the case when all the processes are one-dimensional. In order to solve the optimization problem (2.12), we define the Lagrangian as follows for a given constant λ > 0: J ( CT ,u,v;λ )= E[U2 ( Xu,vT −CT ) + λ ( U1 ( CT ,G u,v T )−R)]. (2.15) Because of our assumptions, by the standard optimization theory (see Luenberger [8]), we have V2 = sup CT ,u,v J ( CT ,u,v; λ ) (2.16) for some λ > 0. Moreover, if the maximum is attained in (2.12) by (CT , u, v), then it is attained by the same triple in the right-hand side of (2.16), and we have E [ U1 ( CT ,G u,v T )]= R. (2.17) Conversely, if there exists λ > 0 and (CT , u, v) such that the maximum is attained in the right-hand side of (2.16) and such that E[U1(CT ,G u,v T )]= R, then (CT , u, v) is also optimal for the problem V2 of (2.12). 2.2. Necessary conditions for optimality. We cannot directly apply standard approaches to deriving necessary cond... |

431 |
The economic theory of agency: The principals problem", American Economic Review
- Ross
- 1973
(Show Context)
Citation Context ...e of admissible contracts. One natural thing to do is to see whether we can have optimal contracts in the option-like form F (XT ), for some deterministic function F , as, for example, in Ross (1973) =-=[13]-=-. In Cadenillas et al. (2004) [1], sufficient conditions 15are found under which there is a contract of such a form which attains the maximal possible utility for the principal, although it does not ... |

327 |
Aggregation and Linearity in the Provision of Intertemporal Incentives.
- Holmstrom, Milgrom
- 1987
(Show Context)
Citation Context ...s simple as possible contracts which induce the agent to implement actions which will lead to the principal attaining the first-best utility. In this paper, we consider principal-agent problems in continuous time, in which both the volatility (the diffusion coefficient) and the drift of the underlying process can Hindawi Publishing Corporation Journal of Applied Mathematics and Stochastic Analysis Volume 2006, Article ID 95203, Pages 1–27 DOI 10.1155/JAMSA/2006/95203 2 Optimal contracts be controlled by the agent. The pioneering paper in the continuous-time framework is Holmstrom and Milgrom [5], which showed that if both the principal and the agent have exponential utilities, then the optimal contract is linear. Their framework, however, is not that of risk-sharing, but of so-called “hidden information” or “moral hazard” case: the principal cannot directly observe the actions of the agent, who controls the drift only. We consider the (harder) hidden information case in a follow-up paper, by Cvitanic et al. [2], in more general models. Here, as mentioned above, we study the full information case, in which the principal can observe the agent’s actions. Although often less realistic t... |

288 |
Aggregation and Linearity
- Holmstrom, Milgrom
- 1985
(Show Context)
Citation Context ...he volatility (the diffusion coefficient) and the drift of the underlying process can be controlled by the agent. The pioneering paper in the continuous-time framework is Holmstrom and Milgrom (1987) =-=[5]-=-, which showed that if both the principal and the agent have exponential utilities, then the optimal contract is linear. Their framework, however, is not that of risk-sharing, but of so-called ”hidden... |

237 | Linear forward-backward stochastic differential equations
- Yong
- 1999
(Show Context)
Citation Context ...heless, we can find optimal contracts in many examples. The stochastic maximum principle is covered in the book Yong and Zhou (1999) [15], while FBSDEs are studied in the monograph Ma and Yong (1999) =-=[8]-=-. The paper is organized as follows: In Section 2 we set up the contracting problem with full information and we find the first-best solution, the one that corresponds to the best controls from the pr... |

205 |
Applied stochastic control of jump diffusions
- Øksendal, Sulem
- 2007
(Show Context)
Citation Context ...ork by using the ”stochastic maximum principle” method of stochastic control theory. (For other applications of stochastic maximum principle in finance, see the recent book by Oksendal and Sulem 2004 =-=[11]-=-.) In general, it is more straightforward to find explicit solutions (when they exist) from the characterization we obtain, compared to the above mentioned methods. We do not discuss the existence of ... |

73 |
Optimal contracts in a continuous-time delegated portfolio management problem,
- Ou-Yang
- 2000
(Show Context)
Citation Context ...specific benchmark portfolio. It can also be interpreted, in the firm context, as a contract in which the principal sells the firm to the agent in exchange for a specific random payment at a given future time. Literature on the first-best case in continuous time includes Muller [10, 11], who finds the solution in the exponential utilities case, when the drift is controlled, and shows how it can be approximated by control revisions taking place at discrete times. Very general framework with several agents and recursive utilities is considered in Duffie et al. [3] and Dumas et al. [4]. Ou-Yang [13] also considers the principal-agent problem in the context of delegated portfolio management. In his paper, the agent controls the volatility and the drift simultaneously. While he restricts the family of allowable contracts, motivated by the fact that the principal may not observe full information, the restricted solution of his problem turns out to be the same as the solution of our full information problem, and thus the restriction does not really matter. That paper uses Hamilton-JacobiBellman equations as the technical tool. In Cadenillas et al. [1] the results of Ou-Yang [13] have been ge... |

60 |
Solution of forward-backward stochastic differential equations
- Hu, Peng
(Show Context)
Citation Context ...ods. We do not discuss the existence of the optimal control. Instead, the stochastic maximum principle enables us to characterize the optimal contract via a solution to forwardbackward stochastic differential equations (FBSDEs), possibly fully coupled. For some of these there is a theory that guarantees existence. However, in general, it is not known under which general conditions these equations have a solution. Nevertheless, we can find optimal contracts in many examples. The stochastic maximum principle is covered in the book by Yong and Zhou [17], while FBSDEs are studied in, for example, [6, 14], and in the monograph by Ma and Yong [9] which contains more references. The paper is organized as follows: in Section 2 we set up the contracting problem with full information and we find the first-best solution, the one that corresponds to the best controls from the principal’s point of view. In Section 3 we show that those controls are implementable, that is, there is a contract which induces the agent to implement the firstbest controls. We present some examples in Section 4. We conclude in Section 5, mentioning possible further research topics. 2. The first-best solution Because we assum... |

50 | Fully coupled forward-backward stochastic differential equations and applications to optimal control
- Peng, Wu
- 1999
(Show Context)
Citation Context ...ods. We do not discuss the existence of the optimal control. Instead, the stochastic maximum principle enables us to characterize the optimal contract via a solution to forwardbackward stochastic differential equations (FBSDEs), possibly fully coupled. For some of these there is a theory that guarantees existence. However, in general, it is not known under which general conditions these equations have a solution. Nevertheless, we can find optimal contracts in many examples. The stochastic maximum principle is covered in the book by Yong and Zhou [17], while FBSDEs are studied in, for example, [6, 14], and in the monograph by Ma and Yong [9] which contains more references. The paper is organized as follows: in Section 2 we set up the contracting problem with full information and we find the first-best solution, the one that corresponds to the best controls from the principal’s point of view. In Section 3 we show that those controls are implementable, that is, there is a contract which induces the agent to implement the firstbest controls. We present some examples in Section 4. We conclude in Section 5, mentioning possible further research topics. 2. The first-best solution Because we assum... |

27 |
Optimization by vector spaces methods. Wiley-Interscience
- Luenberger
- 1997
(Show Context)
Citation Context ...propriate Lagrange multiplier ˆλ, if it exists. We now describe the usual way for identifying it, without giving assumptions for this method to work. Instead, we refer the reader to Luenberger (1969) =-=[7]-=-. First, define ˜V (λ) = sup E[U2(X C,u,v u,v T − CT ) + λU1(CT , G u,v T )] . Then, the appropriate ˆ λ is the one that minimizes ˜ V (λ) − λR, if it exists. If, for this λ = ˆ λ there exists an opti... |

20 | Efficient Intertemporal Allocations with Recursive Utility.”
- Dumas, Uppal, et al.
- 2000
(Show Context)
Citation Context ...outcome of a specific benchmark portfolio. It can also be interpreted, in the firm context, as a contract in which the principal sells the firm to the agent in exchange for a specific random payment at a given future time. Literature on the first-best case in continuous time includes Muller [10, 11], who finds the solution in the exponential utilities case, when the drift is controlled, and shows how it can be approximated by control revisions taking place at discrete times. Very general framework with several agents and recursive utilities is considered in Duffie et al. [3] and Dumas et al. [4]. Ou-Yang [13] also considers the principal-agent problem in the context of delegated portfolio management. In his paper, the agent controls the volatility and the drift simultaneously. While he restricts the family of allowable contracts, motivated by the fact that the principal may not observe full information, the restricted solution of his problem turns out to be the same as the solution of our full information problem, and thus the restriction does not really matter. That paper uses Hamilton-JacobiBellman equations as the technical tool. In Cadenillas et al. [1] the results of Ou-Yang [13... |

15 | Efficient and equilibrium allocations with stochastic differential utility,”
- Duffie, Geoffard, et al.
- 1994
(Show Context)
Citation Context ...yment to be a random outcome of a specific benchmark portfolio. It can also be interpreted, in the firm context, as a contract in which the principal sells the firm to the agent in exchange for a specific random payment at a given future time. Literature on the first-best case in continuous time includes Muller [10, 11], who finds the solution in the exponential utilities case, when the drift is controlled, and shows how it can be approximated by control revisions taking place at discrete times. Very general framework with several agents and recursive utilities is considered in Duffie et al. [3] and Dumas et al. [4]. Ou-Yang [13] also considers the principal-agent problem in the context of delegated portfolio management. In his paper, the agent controls the volatility and the drift simultaneously. While he restricts the family of allowable contracts, motivated by the fact that the principal may not observe full information, the restricted solution of his problem turns out to be the same as the solution of our full information problem, and thus the restriction does not really matter. That paper uses Hamilton-JacobiBellman equations as the technical tool. In Cadenillas et al. [1] the r... |

12 | The first-best sharing rule in the continuous-time principal agent problem with exponential utility,
- Muller
- 1998
(Show Context)
Citation Context ...tween the terminal value of the underlying process and a stochastic, state-contingent benchmark. This should be of significant interest in financial economics, because it justifies the use of linear contracts (paying “shares” rather than “options”), as long as we allow the remaining payment to be a random outcome of a specific benchmark portfolio. It can also be interpreted, in the firm context, as a contract in which the principal sells the firm to the agent in exchange for a specific random payment at a given future time. Literature on the first-best case in continuous time includes Muller [10, 11], who finds the solution in the exponential utilities case, when the drift is controlled, and shows how it can be approximated by control revisions taking place at discrete times. Very general framework with several agents and recursive utilities is considered in Duffie et al. [3] and Dumas et al. [4]. Ou-Yang [13] also considers the principal-agent problem in the context of delegated portfolio management. In his paper, the agent controls the volatility and the drift simultaneously. While he restricts the family of allowable contracts, motivated by the fact that the principal may not observe f... |

12 | Asymptotic efficiency in Dynamic Principal-Agent Problems - Muller - 2000 |

9 | Continuous-time principal-agent problems with hidden action: the weak formulation, Working paper,
- Cvitanic, Wan, et al.
- 2005
(Show Context)
Citation Context ...95203, Pages 1–27 DOI 10.1155/JAMSA/2006/95203 2 Optimal contracts be controlled by the agent. The pioneering paper in the continuous-time framework is Holmstrom and Milgrom [5], which showed that if both the principal and the agent have exponential utilities, then the optimal contract is linear. Their framework, however, is not that of risk-sharing, but of so-called “hidden information” or “moral hazard” case: the principal cannot directly observe the actions of the agent, who controls the drift only. We consider the (harder) hidden information case in a follow-up paper, by Cvitanic et al. [2], in more general models. Here, as mentioned above, we study the full information case, in which the principal can observe the agent’s actions. Although often less realistic than the hidden information case, we would like to point out that our full information framework is directly applicable to the important finance problem of optimal reward of portfolio managers. We will see that in such a context the optimal contracts do not require observing the manager’s actions anyway. We would also like to point out that the extension to continuous-time models is important for these reasons: such a mode... |

7 |
Dynamic Principal-Agent problems with perfect information”. Working paper
- Cadenillas, Cvitanić, et al.
- 2003
(Show Context)
Citation Context ...f our full information problem, and thus the restriction does not really matter. That article uses Hamilton-Jacobi-Bellman equations as the technical tool. In Cadenillas, Cvitanić and Zapatero (2003) =-=[1]-=- the results of Ou-Yang (2003) [12] have been generalized to a setting where the drift is also controlled by the agent independently of the volatility, and the principal observes it. They use duality-... |

5 |
Completeness of security markets and solvability of linear backward stochastic dierential equations, Working paper,
- Yong
- 2004
(Show Context)
Citation Context ...nomial growth, the uniform integrability in (A3)(ii) automatically holds true. If they have exponential growth, there are some discussions on the integrability of exponential processes in Yong (2004) =-=[14]-=-. Note that (2.1) has a unique strong solution: by (A3) (i) we have { E ∫ T [v 2 t + f(t, 0, ut, vt) 2 } ]dt < ∞ 0 Then by boundedness of |fx| and by standard arguments we get E{sup |Xt| t 2 } < ∞. We... |

4 |
Optimal portfolio delegation when parties have different coefficients of risk aversion,
- Larsen
- 2005
(Show Context)
Citation Context ...ory. Because of the limitations of that approach, the results of those two papers are obtained in the setting of linear dynamics (although the cost function is allowed to be nonlinear). Larsen (2005) =-=[6]-=- solves numerically the case with power utilities for the linear, portfolio delegation case, for contracts which depend only on the final value of the portfolio. While this and other existing literatu... |