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, for which the agent's optimal effort is straightforward to find. We then show that the optimization over the restricted family of contracts represents no loss of generality. As a consequence, we have reduced this non-zero sum stochastic differential game to a stochastic control problem which may be addressed by the 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.

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimisation, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalisations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semi-martingale characterisation of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [86]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of [71], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).1 generally based on the stability of the controls with respect to conditioning and concatenation, together with a measurable selection argument, which, roughly speaking, allow to prove the measurability of the associated value function, as well as constructing almost optimal controls through "pasting". This is exactly the approach followed by Bertsekas and Shreve [6], and Dellacherie [25] for discrete time stochastic control problems. In continuous time, a comprehensive study of the dynamic programming principle remained more elusive. Thus, El Karoui,in [34], established the dynamic programming principle for the optimal stopping problem in a continuous time setting, using crucially the strong stability properties of stopping times, as well as the fact that the measurable selection argument can be avoided in this context, since an essential supremum over stopping times can be approximated by a supremum over a countable family of random variables. Later, for general controlled Markov processes (in continuous time) problems, El Karoui, Huu Nguyen and Jeanblanc [36] provided a framework to derive the dynamic programming principle using the measurable selection theorem, by interpreting the controls as probability measures on the canonical trajectory space (see e.g. Theorems 6.2, 6.3 and 6.4 of [36]). Another commonly used approach to derive the DPP was to bypass the measurable selection argument by proving, under additional assumptions, a priori regularity of the value function. This was the strategy adopted, among others, by Fleming and Soner [43], and in the so-called weak DPP of Bouchard and Touzi [15], which has then been extended by Bouchard and Nutz [10,12] and Bouchard, Moreau and Nutz [9] to optimal control problems wit...

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In this paper, we investigate a moral hazard problem in finite time with lump-sum and continuous payments, involving infinitely many Agents with mean field type interactions, hired by one Principal. By reinterpreting the mean-field game faced by each Agent in terms of a mean field forward backward stochastic differential equation (FBSDE for short), we are able to rewrite the Principal's problem as a control problem of McKean-Vlasov SDEs. We review one general approaches to tackle it, introduced recently in [1,43,44,45,46] using dynamic programming and Hamilton-Jacobi-Bellman (HJB for short) equations, and mention a second one based on the stochastic Pontryagin maximum principle, which follows [10]. We solve completely and explicitly the problem in special cases, going beyond the usual linear-quadratic framework. We finally show in our examples that the optimal contract in the N −players' model converges to the mean-field optimal contract when the number of agents goes to +∞, thus illustrating in our specific setting the general results of [8].

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete-time approximations of general martingales.2010 Mathematics Subject Classification. 60G48, 60G55, 60G57, 60H05. Key words and phrases. BSDEs, processes with jumps, stochastically discontinuous martingales, random time horizon, stochastic Lipschitz generator. We thank Martin Schweizer, two anonymous referees and the associate editor for their comments that have resulted in a significant improvement of the manuscript. Alexandros Saplaouras gratefully acknowledges the financial support from the DFG Research Training Group 1845 "Stochastic Analysis with Applications in Biology, Finance and Physics". Dylan Possamaï gratefully acknowledges the financial support of the ANR project PACMAN, ANR-16-CE05-0027. Moreover, all authors gratefully acknowledge the financial support from the PROCOPE project "Financial markets in transition: mathematical models and challenges". 1 The authors are indebted to Saïd Hamadène for pointing out this reference. The published version of [51] states that the article was received on October 27, 1971. It is also present in the bibliography of [21], though it is never referred to in the text. 2 We emphasize that the references given below are just the tip of the iceberg, though most of them are, in our view, among the major ones of the field. Nonetheless, we do not make any claim about comprehensiveness of the following list. 1 2 A. PAPAPANTOLEON, D. POSSAMAÏ, AND A. SAPLAOURAS mention El Karoui and Huang [60], Bender and Kohlmann [18], Wang, Ran and Chen [138] as well as Briand and Confortola [27]. The first results going beyond the linear growth assumption in z, which assumed quadratic growth, were obtained independently by Kobylanski [97, 98, 99] and Dermoune, Hamadène and Ouknine [56], for bounded ξ and f Lipschitz in y. These results were then further studied by Eddhabi and Ouknine [59], and improved by Lepeltier and San Martín [107, 108], Briand, Lepeltier and San Martín [35] and revisited by Briand and Élie [31], but still for bounded ξ. Wellposedness in the quadratic case when ξ has sufficiently large exponential moments was then investigated by Briand and Hu [33, 34], followed by Delbaen, Hu and Richou [54, 55], Essaky and Hassani [66], and Briand and Richou [36]. A specific quadratic setting with only square integrable terminal conditions has been considered recently by Bahlali, Eddahbi and Ouknine [6, 7], while a result with logarithmic growth was also obtained by Bahlali and El Asri [8], and Bahlali, Kebiri, Khelfallah and Moussaoui [13]. The ca...

In this article, we build upon the work of Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190] to define a notion of a second order backward stochastic differential equation reflected on a lower c\`adl\`ag obstacle. We prove existence and uniqueness of the solution under a Lipschitz-type assumption on the generator, and we investigate some links between our reflected 2BSDEs and nonclassical optimal stopping problems. Finally, we show that reflected 2BSDEs provide a super-hedging price for American options in a market with volatility uncertainty.Comment: Published in at http://dx.doi.org/10.1214/12-AAP906 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1003.6053 by other author

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is nondominated. We propose studying this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power, and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally, several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models.KEY WORDS: second-order backward stochastic differential equation, quadratic growth, robust utility maximization, volatility uncertainty.The authors would like to thank two anonymous referees and the associate editor for their helpful remarks and comments.

We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. We identify a family of admissible contracts for which the optimal agent's action is explicitly characterized, and, using the recent theory of singular changes of measures for Itô processes, we study how restrictive this family is. In particular, in the special case of the standard Homlstrom-Milgrom model with fixed volatility, the family includes all possible contracts. We solve the principal-agent problem in the case of CARA preferences, and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the quadratic variation component of the optimal contract.