Skip to main content
SpringerLink
Log in

Dynamic programming approach to principal–agent problems

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. For a recent different approach, see Evans et al. [19]. For each possible agent’s control process, they characterize contracts that are incentive compatible for it. However, their setup is less general than ours, and it does not allow volatility control, for example.

  2. The ℙ-completion of \({\mathbb{G}}\) is defined for any \(t\in[0,T]\) by \(\mathcal{G}^{\mathbb{P}}_{t}:=\sigma(\mathcal{G}_{t}\cup \mathcal{N}^{\mathbb{P}})\), where \(\mathcal{N}^{\mathbb{P}}:=\{A\subseteq\Omega: {A\subseteq} B\text{ with }B\in{\mathcal{F}}_{T}\mbox{, }\mathbb{P}[B]=0\}\).

  3. The ℙ-augmentation of \({\mathbb {G}}\) is defined by \(\mathcal{G}^{\mathbb{P}+}_{T}:=\mathcal{G}^{\mathbb {P}}_{T}\) and for any \(t\in[0,T)\) by \(\mathcal{G}^{\mathbb{P}+}_{t}=\bigcap _{s>t}{\mathcal{G}}_{s}^{\mathbb{P}}\).

  4. The Brownian motion \(W^{\mathbb{M}}\) is defined on a possibly enlarged space if \(\widehat{\sigma}\) is not invertible ℙ-a.s. We refer to Possamaï et al. [32] for the precise statements.

  5. The existing literature on the continuous-time principal–agent problem only addresses the case of drift control, so that admissible controls are described via equivalent probability measures. In our context, we allow volatility control, which in turn implies that our set \(\mathcal{P}\) is not dominated. It is therefore necessary for our approach to have a pathwise definition of stochastic integrals. Notice that the classical pathwise stochastic integration results of Bichteler [4] (see also Karandikar [24]) are not sufficient for our purpose, as we should need to restrict the process \(Z\) to have further pathwise regularity. The recent result of Nutz [29] is perfectly suitable for our context, but it uses the notion of medial limits to define the stochastic integral of any predictable process with respect to any càdlàg semimartingale whose characteristic triplet is absolutely continuous with respect to a fixed reference measure. The existence of medial limits is not guaranteed under the usual set-theoretic framework ZFC (Zermelo–Fraenkel axioms plus the uncountable axiom of choice), and further axioms have to be added. The continuum hypothesis is one among several sufficient axioms for existence of these limits to hold; see [32, Footnote 3] for a further discussion.

  6. Those methods do not work for the general setup of the current paper, which provides a method for principal–agent problems with volatility choice that enables us to solve both the special, first best case of [7], and the second best, moral hazard case; the special case of moral hazard with CARA utility functions and linear output dynamics is solved using the method of this paper in Cvitanić et al. [9].

References

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beneš, V.E.: Existence of optimal stochastic control laws. SIAM J. Control 9, 446–472 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bichteler, K.: Stochastic integration and \(L^{p}\)-theory of semimartingales. Ann. Probab. 9, 49–89 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bolton, P., Dewatripont, M.: Contract Theory. MIT Press, Cambridge (2005)

    Google Scholar 

  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. Cadenillas, A., Cvitanić, J., Zapatero, F.: Optimal risk-sharing with effort and project choice. J. Econ. Theory 133, 403–440 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cvitanić, J., Possamaï, D., Touzi, N.: Moral hazard in dynamic risk management. Manag. Sci. 63, 3328–3346 (2017)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cvitanić, J., Zhang, J.: Contract Theory in Continuous-Time Models. Springer, Berlin (2012)

    MATH  Google Scholar 

  12. Dellacherie, C., Meyer, P.-A.: Probabilities and Potential A: General Theory. North-Holland, New York (1978)

    MATH  Google Scholar 

  13. Ekren, I., Touzi, N., Zhang, J.: Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44, 1212–1253 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ekren, I., Touzi, N., Zhang, J.: Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44, 2507–2553 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. El Karoui, N., Peng, S., Quenez, M.-C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  21. Haussmann, U.G., Lepeltier, J.-P.: On the existence of optimal controls. SIAM J. Control Optim. 28, 851–902 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Holmström, B., Milgrom, P.: Aggregation and linearity in the provision of intertemporal incentives. Econometrica 55, 303–328 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Karandikar, R.L.: On pathwise stochastic integration. In: Stochastic Processes and Their Applications, vol. 57, pp. 11–18 (1995)

    Google Scholar 

  25. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)

    MATH  Google Scholar 

  26. Mastrolia, T., Possamaï, D.: Moral hazard under ambiguity. Preprint (2015). Available online at: arXiv:1511.03616

  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)

    Article  MATH  Google Scholar 

  28. Müller, H.M.: Asymptotic efficiency in dynamic principal–agent problems. J. Econ. Theory 91, 292–301 (2000)

    Article  MATH  Google Scholar 

  29. Nutz, M.: Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17, 1–7 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nutz, M., van Handel, R.: Constructing sublinear expectations on path space. Stoch. Process. Appl. 123, 3100–3121 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Sannikov, Y.: A continuous-time version of the principal–agent problem. Rev. Econ. Stud. 75, 957–984 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Soner, H.M., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67), 1844–1879 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Soner, H.M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields 153, 149–190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Spear, S.E., Srivastava, S.: On repeated moral hazard with discounting. Rev. Econ. Stud. 54, 599–617 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  41. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  42. Sung, J.: Linearity with project selection and controllable diffusion rate in continuous-time principal–agent problems. Rand J. Econ. 26, 720–743 (1995)

    Article  Google Scholar 

  43. Sung, J.: Corporate insurance and managerial incentives. J. Econ. Theory 74, 297–332 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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

Download references

Author information

Authors and Affiliations

  1. Caltech, Humanities and Social Sciences, M/C 228-77, 1200 E. California Blvd., Pasadena, CA, 91125, USA

    Jakša Cvitanić

  2. PSL Research University, CNRS, CEREMADE, Université Paris–Dauphine, 75016, Paris, France

    Dylan Possamaï

  3. CMAP, École Polytechnique, Route de Saclay, 91128, Palaiseau, France

    Nizar Touzi

Authors
  1. Jakša Cvitanić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dylan Possamaï
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nizar Touzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jakša Cvitanić.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cvitanić, J., Possamaï, D. & Touzi, N. Dynamic programming approach to principal–agent problems. Finance Stoch 22, 1–37 (2018). https://doi.org/10.1007/s00780-017-0344-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-017-0344-4

Keywords

Mathematics Subject Classification (2010)

JEL Classification

Search

Navigation