In this talk, we introduce two problems of contract theory, in continuous-time, with a multitude of agents. First, we will study a model of optimal contracting in a hierarchy, which generalises the one-period framework of Sung (2015). The hierarchy is modelled by a series of interlinked principal-agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal (she) can contract with a manager (he), to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts the agents below him. We will see through a simple example that, while the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Therefore, even this relatively simple introductory example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory on 2BSDEs. 

This will lead us to introduce the second problem, namely optimal contracting for demand-response management, which consists in extending the model by Aid, Possamai, and Touzi (2022) to a mean-field of consumers. More precisely, the principal (an electricity producer, or provider) contracts with a continuum of agents (the consumers), to incentivise them to decrease the mean and the volatility of their energy consumption during high peak demand. In addition, we introduce a common noise, impacting all consumption processes, to take into account the impact of weather conditions on the agents’ electricity consumption. This mean-field framework with common noise leads us to consider a more extensive class of contracts. In particular, we prove that the results of [1] can be improved by indexing the contracts on the consumption of one agent and aggregate consumption statistics from the distribution of the entire population of consumers. 

Talk based on Hubert (2020) and Elie, Hubert, Mastrolia, and Possamai (2021).

I will show that a zero-sum stopping game in continuous time with partial and/or asymmetric information admits a saddle point (and, consequently, a value) in randomised stopping times when stopping payoffs of players are general càdlàg adapted processes. We do not assume a Markovian nature of the game nor a particular structure of the information available to the players. I will discuss links with classical results by Baxter, Chacon (1977) and Meyer (1978) derived for optimal stopping problems. Based on a joint work with Tiziano De Angelis, Nikita Merkulov and Jacob Smith.

We present a stochastic Tikhonov theorem for two-scales systems of SDEs, which cover the case of McKean-Vlasov SDEs. Our approach extends and generalizes previous results on two-scales systems of SDEs without mean-field interaction. As an application we provide a novel method for approximating the solution of certain systems of FBSDEs, related to the Pontryagin maximum principle, which is new even for the case without mean-field interaction. This is a joint work with A. Cosso.

We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options and we compute their non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, we find explicitly a family of portfolios which are asymptotically optimal.

In this talk, we introduce several functional law of large numbers (FLLN) and functional central limit theorem (FCLT) for quasi-stationary Hawkes processes. Under some divergence conditions on triggered events, We prove that the normalized point processes can be approximated in distribution by a long-range dependent Gaussian process. Differently, both FLLN and FCLT fail when triggered events satisfy aggregation conditions. In this case, we prove that the rescale Hawkes process converges weakly to the integral of a critical branching diffusion with immigration. Also, the convergence rate is deduced in terms of Fourier-Laplace distance bound and Wasserstein distance bound. This talk is based on a joint work with Ulrich Horst.

This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by Carmona and Lacker, we consider a fully non-Markovian setting allowing for drift control, and interactions through the joint distribution of players’ states and controls. We provide first a new characterisation of mean-field equilibria as arising from solutions to a novel kind of McKean–Vlasov back- ward stochastic differential equations, for which we provide a well-posedness theory. We incidentally obtain there unusual existence and uniqueness results for mean-field equilibria, which do not require short-time horizon, separability assumptions on the coefficients, nor Lasry and Lions’s monotonicity conditions, but rather smallness conditions on the terminal reward. We then take advantage of this characterisation to provide non-asymptotic rates of convergence for the value functions and the Nash-equilibria of the N-player version to their mean-field counterparts, for general open-loop equilibria. This relies on new back- ward propagation of chaos results, which are of independent interest. This is a joint work with Ludovic Tangpi. 

The theory and applications of mean-field games/control have stimulated a growing interest and generated important literature over the last decade since the seminal papers by Lasry/Lions and Caines, Huang, Malhamé. This talk will address some learning methods for numerically solving  continuous time mean-field control problems, also called McKean-Vlasov control (MKV). In a first part, we consider a model-based setting, and present  numerical approximation methods for the Master Bellman equation that characterises the solution to MKV, based on the one hand on particles approximation of the Master equation, and on the other hand on cylindrical neural networks approximation of functions defined on the Wasserstein space. The second part of the lecture is devoted to  a model-free setting, a.k.a. reinforcement learning. We develop a policy gradient approach under entropy regularisation based on a suitable representation of  the gradient value function with respect to parametrised randomised policies. This study leads to actor-critic algorithms for learning simultaneously and alternately value function and optimal policies. Numerical examples in a linear-quadratic mean-field setting illustrate our results. 

We address the problem of Moral Hazard in continuous time between a Principal and an Agent that has time-inconsistent preferences. Building upon previous results on non-Markovian time-inconsistent control for sophisticated agents, we are able to reduce the problem of the principal to a novel class of control problems, whose structure is intimately linked to the representation of the problem of the Agent via a so-called extended Backward Stochastic Volterra Integral equation. We will present some results on the characterization of the solution to problem for different specifications of preferences for both the Principal and the Agent.

Market-wide trading halts, also called circuit breakers, have been widely adopted as part of the stock market architecture, in the hope of stabilizing the market during dramatic price declines.  We develop an intertemporal equilibrium model to examine how circuit breakers impact market behavior and welfare.  We show that a circuit breaker tends to lower the level of price and significantly alters its dynamics.  In particular, as the price approaches the circuit breaker, its volatility rises drastically, accelerating the chance of triggering the circuit breaker -- the so-called ``magnet effect''.  In addition, returns exhibit increasing negative skewness and positive drift, while trading activity spikes up.  Our empirical analysis finds supportive evidence for the model's predictions.  Moreover, we show that a circuit breaker can affect the overall welfare either negatively or positively, depending on the relative significance of investors' trading motives for risk sharing vs. irrational speculation. This is a joint work with Hui Chen, Anton Petukhov, and Jiang Wang.