Send via e-MailImprimer × Table des matières Mathématiques financières et actuarielles, probabilités numériques Prochaines séances Séances passées Groupe de travail Équipe thématique Mathématiques financières et actuarielles, probabilités numériques Gestion des séances Mathématiques financières et actuarielles, probabilités numériques Jour, heure et lieu Le Jeudi à 16:00, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016 Contact(s) Jean-Francois Chassagneux Stephane Crepey Idris Kharroubi Gilles Pages Prochaines séances Mathématiques financières et actuarielles, probabilités numériques Jeudi 30 mars 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209 Mehdi Talbi (ETH Zurich) Non encore annoncé. Mathématiques financières et actuarielles, probabilités numériques Jeudi 6 avril 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209 Mahmoud Khabou (INSA Toulouse) The nonlinear discrete-time Hawkes process The nonlinear Hawkes process is a point process for which the occurrence of future events depends on its history, either by excitation or inhibition. This property made it popular in many fields, such as neuro-sciences and social dynamics. We propose a tractable nonlinear Poisson autoregression as a discrete-time Hawkes process. Our model allows for cross-excitation and inhibition between components, as well as for exogenous random noise on the intensity. We then prove a convergence theorem as the time step goes to zero. Finally, we suggest a parametric calibration method for the continuous-time Hawkes process based on the regression on the discrete-time approximation. Calendrier des séances Pour ajouter le calendrier des séances à votre agenda favori, souscrire au calendrier en indiquant ce lien. Séances passées Année 2023 Mathématiques financières et actuarielles, probabilités numériques Jeudi 23 mars 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209 Ziad Kobeissi (INRIA ILB) Temporal Difference Learning with Continuous Time and State in the Stochastic Setting Classical methods in reinforcement learning allow to efficiently approximate discrete-time value functions using observations. In this talk, we investigate the extensions of such methods to the continuous-time case in a stochastic regime. Mathematically, this boils down to numerically solving BSDEs through learning from observations. Here, we focus our attention on the problem of policy evaluation and more precisely on the basic method TD(0) (for temporal difference). This corresponds to learning the value function of an uncontrolled SDE associated with a reward function. Using vanishing time steps, we propose two adaptations of the TD(0) algorithm: the first is similar to the one in discrete time and is model-free; the second is model-based and is obtained by adding a zero-expectation term, resulting in a reduction of its variance. In the linear setting, we prove multiple convergence results for the two algorithms, the model-based one is more flexible and enjoys better convergence rates. In particular, using the Polyak-Juditsky averaging method and a constant learning step, we obtain a convergence rate similar to the state of the art on the simpler problem of linear regression using SGD. Finally, we present simulations showing numerical evidence of our theoretical analysis. Mathématiques financières et actuarielles, probabilités numériques Jeudi 16 mars 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209 Rudy Morel (Ecole Normale Supérieure) A statistical model of financial time-series through Scattering Spectra We introduce the wavelet scattering spectra which provide non-Gaussian models of time-series having stationary increments. These spectra are an extension of the standard wavelet spectrum and are defined as the diagonal of a certain non-linear correlation matrix on wavelet coefficients. They characterize a wide range of non-Gaussian properties of multi-scale processes. This is analyzed for a variety of processes in the Finance literature. We prove that self-similar processes have scattering spectra which are scale invariant. This property can be tested statistically on a single realization and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering spectra coefficients, and generate new time-series with a microcanonical sampling algorithm. Besides capturing statistical properties of observed time-series, these models can be used to predict future volatility and are shown to capture non-trivial statistical properties of the option smile. Mathématiques financières et actuarielles, probabilités numériques Jeudi 9 mars 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209 Ofelia Bonesini (Imperial College) Correlated equilibria for mean field games with progressive strategies In a discrete space and time framework, we study the mean field game limit for a class of symmetric N-player games based on the notion of correlated equilibrium. We give a definition of correlated solution that allows to construct approximate N-player correlated equilibria that are robust with respect to progressive deviations. We illustrate our definition by way of an example with explicit solutions. Mathématiques financières et actuarielles, probabilités numériques Jeudi 16 février 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209 William Hammersley (Univ. Nice) A prospective regularising common noise for mean field systems Mathématiques financières et actuarielles, probabilités numériques Jeudi 2 février 2023, 16 heures, Jussieu, Salle Paul Lévy, 16-26 209 Nabil Kazi-Tani (Université de Lorraine) The role of correlation in diffusion control ranking games We study Nash equilibriums in 2-players continuous time stochastic differential games, where the players are allowed to control the diffusion coefficient of their state process. We consider zero-sum ranking games, in the sense that the criteria to optimize only depends on the difference of the two players state processes. We explicitly compute the players optimal strategies, depending on the correlation of the Brownian motions driving the two state equations: in particular, if the correlation coefficient is smaller than some explicit threshold, then the optimal strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the optimal controls are mixed strategies. To characterize these equilibria, we rely on a relaxed formulation of the game, allowing the players to randomize their actions. This is a joint work with Stefan Ankirchner and Julian Wendt (University of Jena). Mathématiques financières et actuarielles, probabilités numériques Jeudi 26 janvier 2023, 16 heures 30, Jussieu, Salle Paul Lévy, 16-26 209 Manal Jakani (Le Mans Université) Approximation of reflected SDEs in time-dependent domains and applications to Generalized BSDEs and PDE in time-dependent domain We consider a class of reflected SDEs in non-smooth convex time-dependent domains. We provide a strong approximation for this type of equations using a sequence of standard SDEs. As a consequence, we obtain an approximation scheme for generalized BSDEs using standard BSDEs. As a by-product, we get an approximation for the solution of a system of PDEs with nonlinear boundary conditions in time-dependent domains. Mathématiques financières et actuarielles, probabilités numériques Jeudi 19 janvier 2023, 16 heures 30, Jussieu, Salle Paul Lévy, 16-26 209 Andrea Mazzon (LMU München) Detecting asset price bubbles using deep learning In this paper we employ deep learning techniques to detect financial asset bubbles by using observed call option prices. The proposed algorithm is widely applicable and model-independent. We test the accuracy of our methodology in numerical experiments within a wide range of models and apply it to market data of tech stocks in order to assess if asset price bubbles are present. Under a given condition on the pricing of call options under asset price bubbles, we provide a theoretical foundation of our approach for positive and continuous stochastic processes. When such a condition is not satisfied, we focus on local volatility models. To this purpose, we give a new necessary and sufficient condition for a process with time-dependent local volatility function to be a strict local martingale. This is a joint work with Francesca Biagini, Lukas Gonon and Thilo Meyer-Brandis. Mathématiques financières et actuarielles, probabilités numériques Jeudi 5 janvier 2023, 16 heures, Jussieu, Salle Paul Lévy, 16-26 209 Ahmed Kebaier (Université d'Evry) The interpolated drift implicit Euler scheme Multilevel Monte Carlo method for pricing Barrier options and applications to the CIR and CEV models Recently, Giles et al. [2019] proved that the efficiency of the Multilevel Monte Carlo (MLMC) method for evaluating Down-and-Out barrier options for a diffusion process (Xt)t∈[0,T] with globally Lipschitz coefficients, can be improved by combining a Brownian bridge technique and a conditional Monte Carlo method provided that the running minimum inft∈[0,T]Xt has a bounded density in the vicinity of the barrier. In the present work, thanks to the Lamperti transformation technique and using a Brownian interpolation of the drift implicit Euler scheme of Alfonsi [2013], we show that the efficiency of the MLMC can be also improved for the evaluation of barrier options for models with non-Lipschitz diffusion coefficients under certain moment constraints. We study two example models: the Cox-Ingersoll-Ross (CIR) and the Constant of Elasticity of Variance (CEV) processes for which we show that the conditions of our theoretical framework are satisfied under certain restrictions on the models parameters. In particular, we develop semi-explicit formulas for the densities of the running minimum and running maximum of both CIR and CEV processes which are of independent interest. Finally, numerical tests are processed to illustrate our results. Mathématiques financières et actuarielles, probabilités numériques Jeudi 5 janvier 2023, 17 heures, Jussieu, Salle Paul Lévy, 16-26 209 Thomas Wagenhofer (TU Berlin) Weak error estimates for rough volatility models We consider a rough volatility model where the volatility is a (smooth) function of a Riemann-Liouville Brownian motion with Hurst parameter H in (0,1/2). When simulating these models, one often uses a discretization of stochastic integrals as an approximation. These integrals can be interpreted as log-stock-prices and in financial applications such as in pricing, the most relevent quantities are expectations of (payoff) functions. Our main result is that moments of these integrals have a weak error rate of order 3H+1/2 if H<1/6 and order 1 otherwise. For this we first derive a moment formula for both the discretization and the true stochastic integral. We then use this formula and properties of Gaussian random variables to prove our main theorems. Furthermore, we show that this convergence rate also holds for slightly more general payoffs and also provide a lower bound. Note that our rate of 3H+1/2 is in stark contrast to the strong error rate which is of order H. Année 2022 Mathématiques financières et actuarielles, probabilités numériques Jeudi 15 décembre 2022, 16 heures, Jussieu, Salle Paul Lévy, 16-26 209 Nizar Touzi (CMAP, Ecole Polytechnique) Arrêt optimal en champ moyen Mathématiques financières et actuarielles, probabilités numériques Jeudi 15 décembre 2022, 17 heures, Jussieu, Salle Paul Lévy, 16-26 209 Olivier Bokanowski (Univ. Paris Cité, LJLL) Neural Networks for First Order HJB Equations We consider a deterministic optimal control problem, in a finite horizon context, and propose deep neural network approximations for Bellman's dynamic programming principle, corresponding also to some first-order Hamilton-Jacobi-Bellman (HJB) equations. This work follows the work of Huré, Pham, Bachouch, and Langrené (SIAM J. Numer. Anal., vol. 59 (1), 2021, pp. 525-557) where algorithms are proposed in a stochastic context. However, we need to develop an entirely new approach in order to deal with the propagation of errors in the deterministic setting, where no diffusion is present in the dynamics. Our analysis gives precise error estimates in an average norm. The algorithms are then illustrated on several academic numerical examples related to front propagations models in the presence of obstacle constraints (modelized by an optimal maximum running cost problem) showing the relevance of the approach for average dimensions (e.g. from $2$ to $8$), in particular in situations where the value functions is non-smooth. This is a joint work with Xavier Warin and Averil Prost. Mathématiques financières et actuarielles, probabilités numériques Jeudi 24 novembre 2022, 16 heures 30, Jussieu, Salle Paul Lévy, 16-26 209 Pierre Bras (LPSM, Sorbonne Université) Total variation convergence of the Euler-Maruyama scheme in small time with unbounded drift We give bounds for the total variation distance between the solution of a stochastic differential equation in $R^d$ and its one step Euler-Maruyama scheme in small time. We show that for small t, the total variation is of order $t^{1/3}$ , and more generally of order $t^{r/(2r+1)}$ if the noise coefficient $\sigma$ of the SDE is elliptic and $C^2r_b$, $r \in N$, using multi-step Richardson-Romberg extrapolation. The Richardson-Romberg extrapolation is a method used in numerical analysis to improve the convergence rates of numerical schemes and relies on a linear combination of Taylor expansions with null coefficients up to some order; we adapt this method to our case for theoretical purposes. Mathématiques financières et actuarielles, probabilités numériques Jeudi 20 octobre 2022, 16 heures, Jussieu, Salle Paul Lévy, 16-26 209 Damien Lamberton (Université Gustave Eiffel) Régularité de la frontière libre d'un problème d'arrêt optimal : une approche probabiliste Dans ce travail en collaboration avec Tiziano De Angelis (de l'Université de Turin), nous proposons une approche probabiliste de la dérivabilité de la frontière libre d'un problème d'arrêt optimal d'une diffusion uni-dimensionnelle. Mathématiques financières et actuarielles, probabilités numériques Jeudi 20 octobre 2022, 17 heures, Jussieu, Salle Paul Lévy, 16-26 209 Aurélien Alfonsi (Ecole des Ponts) Approximation of Stochastic Volterra Equations with kernels of completely monotone type (Joint work with Ahmed Kebaier) In this work, we develop a multifactor approximation for d-dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L2-estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in (0, 1/2), we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model. Mathématiques financières et actuarielles, probabilités numériques Jeudi 6 octobre 2022, 16 heures 30, Jussieu, Salle Paul Lévy, 16-26 209 Michaël Allouche (Ecole Polytechnique) Estimation of extreme quantiles from heavy-tailed distributions with neural networks We propose new parametrizations for neural networks in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between extreme log-quantiles and their neural network approximation is established. The finite sample performances of the non-conditional neural network estimator are compared to other bias-reduced extreme-value competitors on simulated data. The source code is available at https://github.com/michael-allouche/nn-quantile-extrapolation.git. Finally, the conditional neural network estimators are implemented to investigate the behavior of extreme rainfalls as functions of their geographical location in the southern part of France. Mathématiques financières et actuarielles, probabilités numériques Jeudi 7 juillet 2022, 16 heures, Sophie Germain salle 1016 Anthony Reveillac (INSA Toulouse) Malliavin calculus for Hawkes functionals and application to Insurance In this talk we will present the basics of the Malliavin calculus we developed for the Hawkes process. We will recall this mathematical object together with one key representation which is at the core of our analysis. As an application of our theoretical findings, we will present some quantifications of the risk for (re-)Insurance contracts. This talk is based on several joint works with Caroline Hillairet; Lorick Huang; Mahmoud Khabou and Mathieu Rosenbaum. Mathématiques financières et actuarielles, probabilités numériques Jeudi 23 juin 2022, 17 heures, Sophie Germain salle 1016 Boualem Djehiche (KTH Stockholm) On zero-sum Dynkin games of mean field type. I will review recent results on a class of zero-sum game problems of mean-field type which extend the classical zero-sum Dynkin game problems. The results I will highlight include sufficient conditions under which such a game admits a value and a saddle point and a characterization of the value of the game in terms of a specific class of doubly reflected backward stochastic differential equations (BSDEs) of mean-field type. Moreover, a corresponding system of weakly interacting values of zero-sum Dynkin games is introduced and is shown to converge to the value of the zero-sum mean-field Dynkin game. In particular, a propagation of chaos result is derived. This is a joint work with Roxana Dumistrescu. Mathématiques financières et actuarielles, probabilités numériques Jeudi 2 juin 2022, 17 heures, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016 Marcos Lopes De Prado (ADIA) Open problems in Finance Mathématiques financières et actuarielles, probabilités numériques Jeudi 12 mai 2022, 17 heures, Sophie Germain salle 1016 Maximilien Germain (Université Paris Cité, LPSM) A level-set approach to the control of state-constrained McKean-Vlasov equations: application to portfolio selection We consider the control of McKean-Vlasov dynamics (or mean-field control) with probabilistic state constraints. We rely on a level-set approach which provides a representation of the constrained problem in terms of an unconstrained one with exact penalization and running maximum or integral cost. Our work extends (Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp. 2568–2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71 (2015), pp. 125–163) to a mean-field setting. The reformulation as an unconstrained problem is particularly suitable for the numerical resolution of the problem, that is achieved from an extension of a machine learning algorithm from (Carmona, Laurière, arXiv:1908.01613 to appear in Ann. Appl. Prob., 2022). An application focuses on a mean-variance portfolio selection problem with probabilistic constraints on the wealth. We also illustrate our approach for a direct numerical resolution of the primal Markowitz continuous-time problem without relying on duality. Mathématiques financières et actuarielles, probabilités numériques Jeudi 28 avril 2022, 16 heures, Sophie Germain salle 1016 Nabil Khazi-Tani (IECL, université de Lorraine) Non encore annoncé. Mathématiques financières et actuarielles, probabilités numériques Jeudi 17 mars 2022, 16 heures, Sophie Germain salle 1016 Pierre Cardaliaguet (Ceremade, Université Paris-Dauphine) On the convergence rate for the optimal control of McKean-Vlasov dynamics In this talk I will report on a joint work with S. Daudin (Paris Dauphine), J. Jackson (U. Texas) and P. Souganidis (U. Chicago). We are interested in the convergence problem for the optimal control of McKean-Vlasov dynamics, also known as mean field control. We establish an algebraic rate of convergence of the value functions of N-particle stochastic control problems towards the value function of the corresponding McKean-Vlasov problem. This convergence rate is established in the presence of both idiosyncratic and common noise, and in a setting where the value function for the McKean-Vlasov problem need not be smooth. Then I will discuss a quantitative propagation of chaos property for the optimal trajectories of the optimal control of the particle system. It relies on the regularity of the value function of the limit problem on an open and dense subset of the state space. Mathématiques financières et actuarielles, probabilités numériques Jeudi 17 mars 2022, 17 heures, Sophie Germain salle 1016 Haoyang Cao (École Polytechnique) Identifiability in Inverse Reinforcement Learning Inverse reinforcement learning attempts to reconstruct the reward function in a Markov decision problem, using observations of agent actions. As already observed in Russell [1998] the problem is ill-posed, and the reward function is not identifiable, even under the presence of perfect information about optimal behavior. We provide a resolution to this non-identifiability for problems with entropy regularization. For a given environment, we fully characterize the reward functions leading to a given policy and demonstrate that, given demonstrations of actions for the same reward under two distinct discount factors, or under sufficiently different environments, the unobserved reward can be recovered up to a constant. We also give general necessary and sufficient conditions for reconstruction of time-homogeneous rewards on finite horizons, and for action-independent rewards, generalizing recent results of Kim [2021] and Fu [2018]. Mathématiques financières et actuarielles, probabilités numériques Jeudi 3 mars 2022, 17 heures, Sophie Germain salle 1016 Jodi Dianetti (Bielefeld University) Submodular mean field games: Existence and approximation of solutions We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. Our approach also allows to treat submodular mean field games with common noise, as well as mean field games with singular controls, optimal stopping and reflecting boundary conditions Mathématiques financières et actuarielles, probabilités numériques Jeudi 17 février 2022, 16 heures, Sophie Germain salle 1016 David Métivier (CMAP, Ecole Polytechnique) Interpretable hidden Markov model for stochastic weather generation and climate change analysis The challenges raised by climate change force industrials to carefully analyze the resilience of their assets to anticipate future weather conditions. In particular, the estimation of future extreme hydrometeorological events, like the frequency of long-lasting dry spells, is critical for hydropower or nuclear generation. Stochastic Weather Generators (SWG) are essential tools to determine these future risks, as they can quickly sample climate statistics from models. They can be either trained on historical data or from simulated data like expert climate change scenarios. In our work, the SWG described and validated with France historical data is based on a spatial Hidden Markov Model (HMM). It generates correlated multisite rain occurrences and amounts, with special attention to the correct reproduction of the distribution of dry and wet spells. The hidden states are viewed as global climate states, e.g., dry all over France, rainy in the north, etc. The resulting model is fully interpretable. We describe how the model recovers large-scale structures that are compared with North Atlantic Oscillations. The model achieves good performances, specifically in terms of extremes, where for example, statistics of drought at the scale of France are well replicated. The model architecture allows easy integration of other weather variables like temperature. In a last part, we show how the model parameters change when trained on RCP climate scenarios. Mathématiques financières et actuarielles, probabilités numériques Jeudi 17 février 2022, 17 heures, Sophie Germain salle 1016 Sergio Pulido (LaMME, ENSIEE) The rough Heston model with self-exciting jumps We introduce a novel affine Volterra stochastic volatility model by adding jumps with a self-exciting structure to the dynamics of the variance process and the log returns in the well-known rough Heston model. In particular, the variance process is given in terms of the solution of a stochastic affine Volterra equation of convolution type with jumps. Thanks to the affine structure of the model we can provide explicit formulas for the Laplace transforms of quantities like log-prices and their quadratic variation, variance swap rates, and integrated variance. These formulas can be exploited to price options using Fourier inversion techniques. The procedure to get the explicit expressions for these transforms essentially relies on the martingale property of complex-valued exponential processes and comparison results for (deterministic) Riccati-Volterra equations. In addition, when the Brownian component in the spot variance dynamics is equal to zero, we devise an exact and efficient conditional simulation scheme. We illustrate with numerical examples the behavior of the implied volatility smiles for options written on the underlying asset and on the VIX. Joint work with Alessandro Bondi, Giulia Livieri and Simone Scotti. Mathématiques financières et actuarielles, probabilités numériques Jeudi 3 février 2022, 16 heures, Jussieu, Salle Paul Lévy, 16-26 209 Peter Tankov (CREST, ENSAE) Optimal Exploration of an Exhaustible Resource with Stochastic Discoveries The standard Hotelling model assumes that the stock of an exhaustible resource is known. We expand on the model by Arrow and Chang that introduced stochastic discoveries and for the first time completely solve such a model using impulse control. The model has two state variables: the ``proven reserves as well as a finite unexplored area available for exploration with constant marginal cost, resulting in a Poisson process of new discoveries. We prove that a frontier of critical levels of ``proven reserves exists, above which exploration is stopped, and below which it happens at infinite speed. This frontier is increasing in the explored area, and higher ``proven'' reserve levels along this critical threshold are indicative of more scarcity, not less. In our stochastic generalization of Hotelling's rule, price expectations conditional on the current state rise at the rate of interest across exploratory episodes. However, the state-dependent conditional expected path of prices realized prior to exhaustion of the exploratory area rises at a rate lower than the rate of interest, consistent with most empirical tests based on observed price histories. Mathématiques financières et actuarielles, probabilités numériques Jeudi 3 février 2022, 17 heures, Jussieu, Salle Paul Lévy, 16-26 209 Alexandre Pannier (Imperial College, Londres) Rough multi-factor volatility models for SPX and VIX After a short review of VIX and rough volatility models, we provide explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models. Our general setup encompasses both European options on a stock and VIX options, thereby providing new insights on their joint calibration. This framework also allows to consider rough volatility models and their multi-factor versions; in particular we develop a detailed analysis of the two-factor rough Bergomi model. The tools used are essentially based on Malliavin calculus for Gaussian processes. This is a joint work with A. Jacquier and A. Muguruza. Mathématiques financières et actuarielles, probabilités numériques Jeudi 20 janvier 2022, 17 heures, Jussieu, Salle Paul Lévy, 16-26 209 Philippe Bergault (Ecole Polytechnique) A mean field game of market making against strategic traders We design a market making model à la Avellaneda–Stoikov in which the market takers act strategically, in the sense that they design their trading strategy based on a specific signal. The market maker chooses her quotes based on the average market takers' behaviour, modelled through a mean field interaction. We derive, up to the resolution of a coupled HJB–Fokker–Planck system, the optimal controls of the market maker and the representative market taker. This approach is flexible enough to incorporate different behaviours for the market takers and takes into account the impact of their strategies on the price process. Joint work with Bastien Baldacci and Dylan Possamai.