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].

In this paper we study a utility maximization problem with random horizon and reduce it to the analysis of a specific BSDE, which we call BSDE with singular coefficients, when the support of the default time is assumed to be bounded. We prove existence and uniqueness of the solution for the equation under interest. Our results are illustrated by numerical simulations.

No abstract

We address the mechanism design problem of an exchange setting suitable maketake fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system.

In this paper, we extend the Holmström and Milgrom problem [47] by adding uncertainty about the volatility of the output for both the Agent and the Principal. We study more precisely the impact of the "Nature" playing against the Agent and the Principal by choosing the worst possible volatility of the output. We solve the first-best and the second-best problems associated with this framework and we show that optimal contracts are in a class of contracts similar to [14,15], linear with respect to the output and its quadratic variation. We compare our results with the classical problem in [47].to designing optimal incentives, and are therefore present in a very large number of situations. Beyond the obvious application to the optimal remuneration of an employee, one can for instance think on how regulators with imperfect information and limited policy instruments can motivate banks to operate entirely in the social interest, on how a company can optimally compensate its executives, on how banks achieve optimal securitisation of mortgage loans or on how investors should pay their portfolio managers (see Bolton and Dewatripont [6] or Laffont and Martimort [49] for many more examples).Early studies of the risk-sharing problem can be found, among others, in Borch [7], Wilson [99] or Ross [78]. Since then, a large literature has emerged, solving very general risk-sharing problems, for instance in a framework with several Agents and recursive utilities (see Duffie et al. [25] or Dumas et al. [26], or for studying optimal compensation of portfolio managers (see or Cadenillas et al. [11]). From the mathematical point of view, these problems can usually be tackled using either their dual formulation or the so-called stochastic maximum principle, which can characterize the optimal choices of the Principal and the Agent through coupled systems of Forward Backward Stochastic Differential Equations (FBSDEs in the sequel) (see the very nice monograph [20] by Cvitanić and Zhang for a systematic presentation). One of the main findings in almost all of these works, is that one can find an optimal contract which is linear in the terminal value of the output managed by the Agent (a result already obtained in [78]) and possibly some benchmark to which his performance is compared. In specific cases, one can even have Markovian optimal contracts which are given as a (possibly linear) functional of the terminal value of the output (see in particular [11] for details).Concerning the so-called moral hazard problem, the first paper on continuous-time Principal-Agent problems is the seminal paper by Holmström and Milgrom [47]. They consider a Principal and an Agent with exponential utility functions and find that the optimal contract is linear. Their work was generalized by Schättler and Sung [83,84], Sung [90,91], Müller [54,55], and Hellwig and Schmidt [46], using a dynamic programming and martingales approach, which is classical in stochastic control theory (see also the survey paper by Sung [92] for more references). The papers by Wi...

No abstract

In this paper, we study the existence of densities (with respect to the Lebesgue measure) for marginal laws of the solution (Y, Z) to a quadratic growth BSDE. Using the (by now) well-established connection between these equations and their associated semi-linear PDEs, together with the Nourdin-Viens formula, we provide estimates on these densities.

We study the problem of demand response contracts in electricity markets by quantifying the impact of considering a continuum of consumers with mean–field interaction, whose consumption is impacted by a common noise. We formulate the problem as a Principal–Agent problem with moral hazard in which the Principal—she—is an electricity producer who observes continuously the consumption of a continuum of risk‐averse consumers, and designs contracts in order to reduce her production costs. More precisely, the producer incentivizes each consumer to reduce the average and the volatility of his consumption in different usages, without observing the efforts he makes. We prove that the producer can benefit from considering the continuum of consumers by indexing contracts on the consumption of one Agent and aggregate consumption statistics from the distribution of the entire population of consumers. In the case of linear energy valuation, we provide closed‐form expression for this new type of optimal contracts that maximizes the utility of the producer. In most cases, we show that this new type of contracts allows the Principal to choose the risks she wants to bear, and to reduce the problem at hand to an uncorrelated one.