**Beatrice Acciaio**

*Title:* Generalised McKean-Vlasov stochastic control problems.

*Abstract:* I will consider McKean-Vlasov stochastic control problems where the cost functions and the state dynamics depend upon the joint distribution of the controlled state and the control process. First, I will provide a suitable version of the Pontryagin stochastic maximum principle, showing that, in the present general framework, pointwise minimisation of the Hamiltonian with respect to the control is not a necessary optimality condition. Then I will take a different perspective, and present a variational approach to study a weak formulation of such control problems, thereby establishing a new connection between those and optimal transport problems on path space. The talk is based on a joint project with J. Backhoff Veraguas and R. Carmona.

**René Aïd**

*Title:* Optimal trade of flexibility in electricity markets.

*Abstract:* We formulate the problem of demand response contracts in electricity markets as a Principal-Agent problem with moral hazard. The Principal is a risk-averse producer subject to energy generation cost and to variation of generation costs due to limited flexibility. The Principal observes in continuous-time the consumption of the risk-averse Agent who is a single consumer, but he does not observe the efforts she makes to reduce her consumption and the volatility of her consumption in her different usages. Using the dynamic programming methods of Cvitanić, Possamaï and Touzi (2015), we find closed-form expression for the optimal contract that maximises the utility of the principal in the case of linear discrepancy energy valuation. We analyse how the producer shares volatility risk with the consumer and the gain obtained by the producer from the optimal contract. Joint work with Dylan Possamaï and Nizar Touzi.

**Julio Backhoff Veraguas**

*Title:* Risk sensitive optimal transport as a limit of particle systems.

*Abstract:* Optimal transport, a linear programming problem, can be obtained as a limit of entropic optimization problems. The latter, also known as Schrödinger problems, have a solid interpretation in terms of non-interacting particle systems. In this way Optimal transport itself can be given a probabilistic interpretation in terms of random particles in the zero-noise limit. In this talk we introduce a family of (non-linear) risk-sensitive optimal transport problems, and prove that such problems can be interpreted via a zero-noise limit of a system of random particles with random weights. Risk-sensitive optimal transport problems serve to find worst-case dependence structures between random variables, and are therefore relevant in actuarial science and finance. This is work in progress with G. Conforti and M. Beiglböck.

**Jakša Cvitanić**

*Title:* Large tournament games.

*Abstract:* We consider a tournament game in which each player is rewarded based on her rank in terms of the time of reaching a goal. We prove existence, uniqueness and stability of the game with infinitely many players, existence of an approximate equilibrium with finitely many players, and find an explicit characterisation when players are homogeneous. In our setup we find that:

- the welfare may be increasing in cost of effort;
- when the total pie is small, the aggregate effort may be increasing in prize inequality;
- the welfare may go up with a higher percentage of unskilled workers, as do the completion rates of the skilled and unskilled subpopulations.

Our results lend support to government subsidies for R&D, assuming the profits to be made are substantial. (Joint with Erhan Bayraktar and Yuchong Zhang).

**Nicolás Hernández Santibáñez**

*Title:* Nature vs. Agents: moral hazard in a VUCA world.

*Abstract:* In this paper we investigate a problem of contract theory with moral hazard in which the Agent and the Principal play against a third uncertain entity. We extend the seminal framework of Holmström and Milgrom by mixing the natural Stackelberg equilibrium with the worst-case scenario considered, we consider a general controlled model, with general utility functions in the spirit of [14], without needing a dynamic programming principal. We show more exactly that optimal contracts depends on the output and its quadratic variation, as an extension of the works of [32] (by dropping all the restrictive assumptions) and [49] (by reinforcing rigorously the considered form of contracts). We characterise the best reaction effort of the agent through the solution of a second order BSDE and show that the value of the problem of the Principal is the viscosity solution to the corresponding HJBI equation by using the Perron’s method.

**Antoine Jacquier**

*Title:* A uniform approach to pathwise moderate deviations for diffusions.

*Abstract: *Moderate deviations sit half way between central limit theorems and large deviations. Friz, Gerhold and Pinter recently showed that they allowed to determine (small-time) asymptotics of option prices and implied volatilities in regime where the strike depends on the maturity (mimicking actual market behaviour). We extend their result here by considering a pathwise version, which allows us to derive small-time, large-time and tail behaviours altogether. Our results rely on weak convergence techniques for multiscale diffusion processes. Joint work with Konstantinos Spiliopoulos (Boston University).

**Kostas Kardaras**

*Title:* Density of (non)complete market extensions.

*Abstract:* In a market with m liquid assets and n sources of uncertainty, where m<n, we consider the question of whether the set of (non)complete market extensions via options, parametrised by consistent valuation measures, is dense in the set of all possible market extensions. Workable sufficient conditions are given, which illuminate especially the case of continuous filtrations. (This talk is based on joint work with Sergio Pulido).

**Nabil Kazi-Tani**

*Title:* Dependence structures and optimal exchange contracts

*Abstract:* During this talk, we will study the situation where two economic agents seek to exchange part of their respective risks. One motivation for this type of model is the interaction between certain insurers, who are exposed to longevity risk, and some reinsurers who are exposed to mortality risk. We will first show how an optimal exchange problem simplifies to an inf-convolution of the agents risk measures, evaluated at the sum of their risks. In situations where the inf-convolution operation can be computed explicitly, we study the effect of the dependence structure of both risks on the optimal exchange contract. The dependence structure will be modelled by the copula function associated to the vector of risks. This is a joint work with Yahia Salhi (LSAF, Lyon 1 University).

**Daniel Lacker**

*Title:*Local dynamics for large sparse networks of interacting diffusions.

*Abstract:*We study large systems of diffusions (particles) in which each particle is associated with a vertex in a graph and interacts only with its neighbours. The complete graph case is well understood, and the limit as the numbers of particles grows to infinity gives rise to the McKean-Vlasov equation, which describes the behaviour of one typical particle. For general (sparse) graphs, the system is no longer exchangeable, and mean field arguments do not apply. Nevertheless, if the underlying graph converges locally to an infinite regular tree or a Galton-Watson tree, we show that the limiting dynamics of a single particle and its nearest neighbours are described by a peculiar but autonomous system of equations. This work is motivated in part by recent mean field models of inter-bank lending, which capture several features of systemic risk but thus far lack realistic network structure.

**Thibaut Mastrolia**

*Title:*An incentive-based approach for managing market makers’ activities.

*Abstract:*In this work we consider a platform aiming at influencing a market maker to increase the liquidity of a risky asset. We study the impact of such incentive policy on the market and on the profit and loss of each party. Joint work with Omar El Euch, Mathieu Rosenbaum and Nizar Touzi.

**Marcel Nutz**

*Title:* Convergence to the mean-field game limit: a case study.

*Abstract:* We study the convergence and multiplicity of equilibria in a game of optimal stopping. If the mean field game has a unique equilibrium, any sequence of n-player equilibria converges to it as n goes to infinity. Whereas in the case of non-uniqueness, it is shown that an additional stability condition is needed to ensure that a mean field equilibrium is the limit of n-player equilibria. The convergence results are used to discuss the nature of the non-uniqueness in the mean field game. Indeed, in the n-player game, multiplicity is completely explained by groups of agents choosing to coordinate their actions in ways that ensure optimality for the group. The range of equilibria typically converges to the range in the mean field game, thus yielding a clear intuition for the latter. (Joint work with Xiaowei Tan)

**Miquel Oliu Barton**

*Title:* Constant payoff in zero-sum stochastic games

*Abstract:*In any one-shot zero-sum game, the payoff induced by a pair of optimal strategies is equal to the value of the game. For dynamic games, a natural refinement is that the average payoff, after any fraction of the game, is equal to the value of the game. This paper proves that such a property holds in any finite zero-sum stochastic games with patient players, as conjectured by Sorin, Venel and Vigeral (2010). It is a joint work with Bruno Ziliotto (CNRS, Université Paris-Dauphine).

**Athena Picarelli**

*Title:* State constrained optimal control problems via reachability approach.

*Abstract:* This work deals with a class of stochastic optimal control problems in the presence of state constraints. It is well known that for such problems the value function is, in general, discontinuous, and its characterisation by a Hamilton-Jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics of the controlled system. Here, we give a characterisation of the epigraph of the value function without assuming the usual controllability assumptions. To this end, the stochastic optimal control problem is first translated into a state-constrained stochastic target problem. Then a level-set approach is used to describe the backward reachable sets of the new target problem. It turns out that these backward reachable sets describe the value function. The main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalisation. However, the target problem involves a new state variable and a new control variable that is unbounded.

**Teemu Pennanen**

*Title:* Convex duality in nonlinear optimal transport.

*Abstract:* We study problems of optimal transport, by embedding them in a general functional analytic framework of convex optimisation. This provides a unified treatment of a large class of related problems in probability theory and allows for generalisations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the mass transportation problem and its nonlinear generalisations.

**Anthony Réveillac**

*Title:*Pricing formulae for derivatives in insurance using the Malliavin calculus.

*Abstract:*In this talk we will derive a pricing formula using the Malliavin calculus that we apply for the pricing of different classes of derivatives in insurance and for the risk analysis. We will discuss the application of this formula regarding the classical models used in insurance. This talk is based on a joint work with Caroline Hillairet (ENSAE-Paris) and Ying Jiao (ISFA-Lyon).

**Mathieu Rosenbaum**

*Title:* Market impact can only be power law and it implies diffusive prices with rough volatility.

*Abstract:* Market impact is the link between the volume of an order and the price move during and after the execution of this order. We show that under no-arbitrage, the market impact function can only be of power law-type. Furthermore we prove that this implies that the long term price is diffusive with rough volatility. Hence we simply explain the universally rough behaviour of the volatility as a consequence of the no-arbitrage property. From a mathematical viewpoint, our study relies in particular on new existence and uniqueness results for rough Volterra stochastic equations. This is joint work with Paul Jusselin.

**Johannes Ruf**

*Title:* Filtration shrinkage, the structure of deflators, and the failure of market completeness

*Abstract:* We analyse the structure of stochastic discount factors (SDFs) projected on smaller filtrations. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of SDFs. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected SDFs are themselves SDFs in the smaller information market. Finally, we demonstrate that these projections are unable to span all possible SDFs in the smaller information market, by means of an interesting example where market completeness is not retained under filtration shrinkage. This is joint work with Kostas Kardaras.

**Alexandros Saplaouras**

*Title: *The stability property under Mémin’s framework

*Abstract:* In this talk we will discuss initially the stability property of semimartingales, where we refine the result obtained by Mémin. Then we focus on the special case where the sequence of semimartingales consists of solutions of Backward Stochastic Differential Equations with Jumps, in short BSDEJ, and we provide a suitable framework for obtaining the stability property of BSDEJ.

**Nizar Touzi**

*Title:* Random horizon backward SDEs and Principal-Agent problem

*Abstract:* Backward SDEs can be viewed as the path-dependent analogue of Sobolev solutions of parabolic second order PDEs. We provide some recent results in the context of random horizon. This corresponds to an extension of elliptic PDEs. As an application of these results, we provide a systematic method for solving general Principal-Agent problems with possibly infinite horizon. Our main result reduces such Stackelberg stochastic differential games to a standard stochastic control problem, which may be addressed by the standard tools of control theory.

**Chao Zhou**

*Title:*Investment decisions and falling cost of data analytics.

*Abstract:*We study how the cost of data analytics and the characteristics of investors and investment opportunities affect investment decisions and their data analytics. We show that the falling cost of the data analytics raises investors’ leverage, financially constrained or highly risk averse investors use less data analytics, and the demand of data analytics is highest with high expected return opportunities. Due to the increased leverage, the falling cost of data analytics leads to higher losses during the crises. This is a joint work with Jussi Keppo and Hong Ming Tan.