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Dynamic programming approach to principal–agent problems

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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.

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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].

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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
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  2. Dylan Possamaï
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  3. Nizar Touzi
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Correspondence to Jakša Cvitanić.

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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

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