TY - JOUR

T1 - Random Horizon Principal-Agent Problems

AU - Lin, Yiqing

AU - Ren, Zhenjie

AU - Touzi, Nizar

AU - Yang, Junjian

N1 - Funding Information:
\ast Received by the editors February 25, 2020; accepted for publication (in revised form) November 2, 2021; published electronically February 10, 2022. https://doi.org/10.1137/20M1321620 Funding: This work was supported by ERC Advanced Grant 321111 and by the Chairs Financial Risk and Finance and Sustainable Development and the NSFC grant 11801365. \dagger School of Mathematical Sciences, Shanghai Jiao Tong University, 200240 Shanghai, China (yiqing.lin@sjtu.edu.cn). \ddagger CEREMADE, Universit\e' Paris Dauphine, F-75775 Paris Cedex 16, 75016, France (ren@ ceremade.dauphine.fr). \S CMAP, Ecole\' Polytechnique, F-91128 Palaiseau Cedex, France (nizar.touzi@polytechnique.edu). \P FAM, Fakult\a"t fu\"r Mathematik und Geoinformation, Vienna University of Technology, A-1040 Vienna, Austria (junjian.yang@tuwien.ac.at).
Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics Publications. All rights reserved.

PY - 2022

Y1 - 2022

N2 - We consider a general formulation of the random horizon principal-agent problem with a continuous payment and a lump-sum payment at termination. In the European version of the problem, the random horizon is chosen solely by the principal with no other possible action from the agent than exerting effort on the dynamics of the output process. We also consider the American version of the contract, where the agent can also quit by optimally choosing the termination time of the contract. Our main result reduces such nonzero-sum stochastic differential games to appropriate stochastic control problems which may be solved by standard methods of stochastic control theory. This reduction is obtained by following the Sannikov [Rev. Econom. Stud., 75 (2008), pp. 957-984] approach, further developed in [J. Cvitanic, D. Possamai}, and N. Touzi, Finance Stoch., 22 (2018), pp. 1-37]. We first introduce an appropriate class of contracts for which the agent's optimal effort is immediately characterized by the standard verification argument in stochastic control theory. We then show that this class of contracts is dense in an appropriate sense, so that the optimization over this restricted family of contracts represents no loss of generality. The result is obtained by using the recent well-posedness result of random horizon second-order backward SDEs in [Y. Lin, Z. Ren, N. Touzi, and J. Yang, Electron. J. Probab., 25 (2020), 99].

AB - We consider a general formulation of the random horizon principal-agent problem with a continuous payment and a lump-sum payment at termination. In the European version of the problem, the random horizon is chosen solely by the principal with no other possible action from the agent than exerting effort on the dynamics of the output process. We also consider the American version of the contract, where the agent can also quit by optimally choosing the termination time of the contract. Our main result reduces such nonzero-sum stochastic differential games to appropriate stochastic control problems which may be solved by standard methods of stochastic control theory. This reduction is obtained by following the Sannikov [Rev. Econom. Stud., 75 (2008), pp. 957-984] approach, further developed in [J. Cvitanic, D. Possamai}, and N. Touzi, Finance Stoch., 22 (2018), pp. 1-37]. We first introduce an appropriate class of contracts for which the agent's optimal effort is immediately characterized by the standard verification argument in stochastic control theory. We then show that this class of contracts is dense in an appropriate sense, so that the optimization over this restricted family of contracts represents no loss of generality. The result is obtained by using the recent well-posedness result of random horizon second-order backward SDEs in [Y. Lin, Z. Ren, N. Touzi, and J. Yang, Electron. J. Probab., 25 (2020), 99].

KW - first best and second best contracting

KW - moral hazard

KW - random horizon

KW - second-order backward SDE

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U2 - 10.1137/20M1321620

DO - 10.1137/20M1321620

M3 - Article

AN - SCOPUS:85130321591

SN - 0363-0129

VL - 60

SP - 355

EP - 384

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

IS - 1

ER -