Advertisement

Finance and Stochastics

, Volume 22, Issue 1, pp 1–37 | Cite as

Dynamic programming approach to principal–agent problems

  • Jakša CvitanićEmail author
  • Dylan Possamaï
  • Nizar Touzi
Article

Abstract

We consider a general formulation of the principal–agent problem with a lump-sum payment on a finite horizon, providing a systematic method for solving such problems. Our approach is the following. We first find the contract that is optimal among those for which the agent’s value process allows a dynamic programming representation, in which case the agent’s optimal effort is straightforward to find. We then show that the optimization over this restricted family of contracts represents no loss of generality. As a consequence, we have reduced a non-zero-sum stochastic differential game to a stochastic control problem which may be addressed by standard tools of control theory. Our proofs rely on the backward stochastic differential equations approach to non-Markovian stochastic control, and more specifically on the recent extensions to the second order case.

Keywords

Stochastic control of non-Markov systems Hamilton–Jacobi–Bellman equations Second order backward SDEs Principal–agent problem Contract theory 

Mathematics Subject Classification (2010)

91B40 93E20 

JEL Classification

C61 C73 D82 J33 M52 

References

  1. 1.
    Aïd, R., Possamaï, D., Touzi, N.: Electricity demand response and optimal contract theory. Working paper (2017). Available online at: https://sinews.siam.org/Details-Page/electricity-demand-response-and-optimal-contract-theory-2
  2. 2.
    Beneš, V.E.: Existence of optimal strategies based on specified information, for a class of stochastic decision problems. SIAM J. Control 8, 179–188 (1970) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beneš, V.E.: Existence of optimal stochastic control laws. SIAM J. Control 9, 446–472 (1971) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bichteler, K.: Stochastic integration and \(L^{p}\)-theory of semimartingales. Ann. Probab. 9, 49–89 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bolton, P., Dewatripont, M.: Contract Theory. MIT Press, Cambridge (2005) Google Scholar
  6. 6.
    Briand, P., Delyon, B., Hu, Y., Pardoux, É., Stoica, L.: \({L}^{p}\) solutions of backward stochastic differential equations. In: Stochastic Processes and Their Applications, vol. 108, pp. 109–129 (2003) Google Scholar
  7. 7.
    Cadenillas, A., Cvitanić, J., Zapatero, F.: Optimal risk-sharing with effort and project choice. J. Econ. Theory 133, 403–440 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheridito, P., Soner, H.M., Touzi, N., Victoir, N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Commun. Pure Appl. Math. 60, 1081–1110 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cvitanić, J., Possamaï, D., Touzi, N.: Moral hazard in dynamic risk management. Manag. Sci. 63, 3328–3346 (2017) CrossRefGoogle Scholar
  10. 10.
    Cvitanić, J., Wan, X., Zhang, J.: Optimal compensation with hidden action and lump-sum payment in a continuous-time model. Appl. Math. Optim. 59, 99–146 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cvitanić, J., Zhang, J.: Contract Theory in Continuous-Time Models. Springer, Berlin (2012) zbMATHGoogle Scholar
  12. 12.
    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential A: General Theory. North-Holland, New York (1978) zbMATHGoogle Scholar
  13. 13.
    Ekren, I., Touzi, N., Zhang, J.: Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44, 1212–1253 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ekren, I., Touzi, N., Zhang, J.: Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44, 2507–2553 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    El Karoui, N., Huu Nguyen, D., Jeanblanc-Picqué, M.: Compactification methods in the control of degenerate diffusions: existence of an optimal control. Stochastics 20, 169–219 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    El Karoui, N., Peng, S., Quenez, M.-C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    El Karoui, N., Tan, X.: Capacities, measurable selection and dynamic programming. Part I: Abstract framework. Preprint (2013). Available online at: https://arxiv.org/pdf/1310.3363.pdf
  18. 18.
    El Karoui, N., Tan, X.: Capacities, measurable selection and dynamic programming. Part II: Application in stochastic control problems. Preprint (2015). Available online at: arXiv:1310.3364
  19. 19.
    Evans, L.C., Miller, C.W., Yang, I.: Convexity and optimality conditions for continuous time principal–agent problems. Preprint (2015). Available online at: https://math.berkeley.edu/~evans/principal_agent.pdf
  20. 20.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006) zbMATHGoogle Scholar
  21. 21.
    Haussmann, U.G., Lepeltier, J.-P.: On the existence of optimal controls. SIAM J. Control Optim. 28, 851–902 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hellwig, M.F., Schmidt, K.M.: Discrete-time approximations of the Holmström–Milgrom Brownian-motion model of intertemporal incentive provision. Econometrica 70, 2225–2264 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Holmström, B., Milgrom, P.: Aggregation and linearity in the provision of intertemporal incentives. Econometrica 55, 303–328 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Karandikar, R.L.: On pathwise stochastic integration. In: Stochastic Processes and Their Applications, vol. 57, pp. 11–18 (1995) Google Scholar
  25. 25.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991) zbMATHGoogle Scholar
  26. 26.
    Mastrolia, T., Possamaï, D.: Moral hazard under ambiguity. Preprint (2015). Available online at: arXiv:1511.03616
  27. 27.
    Müller, H.M.: The first-best sharing rule in the continuous-time principal–agent problem with exponential utility. J. Econ. Theory 79, 276–280 (1998) CrossRefzbMATHGoogle Scholar
  28. 28.
    Müller, H.M.: Asymptotic efficiency in dynamic principal–agent problems. J. Econ. Theory 91, 292–301 (2000) CrossRefzbMATHGoogle Scholar
  29. 29.
    Nutz, M.: Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17, 1–7 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nutz, M., van Handel, R.: Constructing sublinear expectations on path space. Stoch. Process. Appl. 123, 3100–3121 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Possamaï, D., Tan, X., Zhou, C.: Stochastic control for a class of nonlinear kernels and applications. Annals of Probability (2015), forthcoming. Available online at: arXiv:1510.08439
  33. 33.
    Ren, Z., Touzi, N., Zhang, J.: Comparison of viscosity solutions of semi-linear path-dependent PDEs. Preprint (2014). Available online at: arXiv:1410.7281
  34. 34.
    Sannikov, Y.: A continuous-time version of the principal–agent problem. Rev. Econ. Stud. 75, 957–984 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Schättler, H., Sung, J.: The first-order approach to the continuous-time principal–agent problem with exponential utility. J. Econ. Theory 61, 331–371 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Schättler, H., Sung, J.: On optimal sharing rules in discrete- and continuous-time principal–agent problems with exponential utility. J. Econ. Dyn. Control 21, 551–574 (1997) CrossRefzbMATHGoogle Scholar
  37. 37.
    Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the \({G}\)-expectation. In: Stochastic Processes and Their Applications, vol. 121, pp. 265–287 (2011) Google Scholar
  38. 38.
    Soner, H.M., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67), 1844–1879 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Soner, H.M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields 153, 149–190 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Spear, S.E., Srivastava, S.: On repeated moral hazard with discounting. Rev. Econ. Stud. 54, 599–617 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1997) CrossRefzbMATHGoogle Scholar
  42. 42.
    Sung, J.: Linearity with project selection and controllable diffusion rate in continuous-time principal–agent problems. Rand J. Econ. 26, 720–743 (1995) CrossRefGoogle Scholar
  43. 43.
    Sung, J.: Corporate insurance and managerial incentives. J. Econ. Theory 74, 297–332 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sung, J.: Optimal contracting under mean-volatility ambiguity uncertainties. Preprint (2015). Available online at: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2601174
  45. 45.
    Williams, N.: On dynamic principal–agent problems in continuous time. University of Wisconsin (2009). Preprint. Available online at: http://www.ssc.wisc.edu/~nwilliam/dynamic-pa1.pdf

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Jakša Cvitanić
    • 1
    Email author
  • Dylan Possamaï
    • 2
  • Nizar Touzi
    • 3
  1. 1.CaltechHumanities and Social SciencesPasadenaUSA
  2. 2.PSL Research University, CNRS, CEREMADEUniversité Paris–DauphineParisFrance
  3. 3.CMAPÉcole PolytechniquePalaiseauFrance

Personalised recommendations