2021 Seminars
December 16th, 2021
Speaker: Marcel Nutz (Columbia University)
Title: Stability of Entropic Optimal Transport and Convergence of Sinkhorn’s Algorithm
Abstract: We discuss entropically regularized optimal transport and its stability with respect to the marginals. A qualitative result (for weak convergence) is obtained using the geometric notion of c-cyclical monotonicity and a quantitative result (for Wasserstein distance) is obtained by control theoretic methods. These results can be applied to deduce convergence of Sinkhorn’s algorithm for unbounded cost functions such as the quadratic cost and find a convergence rate in Wasserstein sense. Based on joint works with Espen Bernton, Stephan Eckstein, Promit Ghosal, Johannes Wiesel.
Location and Time: Online via Zoom, 10:00am-11:00am.
November 17th, 2021
Speaker: Zhenjie Ren (Université Paris-Dauphine, France)
Title: Entropic Fictitious Play
Abstract: The classical fictitious play is a common algorithm for solving games. However, once the cost functions of the players are not convex, the method becomes hard to implement. In our study, we add the entropic regulariser, a common strategy for non-convex optimisation, to the cost functions, and look into the analog of fictitious play in this context. We shall further see that the entropic fictitious play not only helps to solve non-convex games, but also can be used to solve optimisations on the space of probability measures, and thus can be applied to train neural networks.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
November 4th, 2021
Speaker: Ying Hu (Université Rennes I, France)
Title: Some Recent Well-Posedness Results on Backward Stochastic Differenctial Equations
Abstract: The aim of this talk is to give some recent well- posedness results for scalar valued Backward Stochastic Differential Equations (BSDEs) when the generator has a sublinear growth, linear growth, superlinear growth, subquadratic growth, quadratic growth and superquadratic growth in the second variable. In each of these six cases, we give some precise conditions for terminal random variables to guarantee the existence and uniqueness of the solution. On the other hand, we give a few results on multi- dimensional BSDEs with quadratic growth whose solvability remains largely open. Finally we mention some perspectives of related research on FBSDE, mean-field FBSDE, G-BSDEs, second-order BSDEs, Volterra BSDEs, etc.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
September 29th, 2021
Speaker: Masaaki Fukasawa (Osaka University, Japan)
Title: Realized Cumulants
Abstract: Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and Friz et al. (2020).
Location and Time: Online via Zoom, 3:30pm-4:30pm.
September 22nd, 2021
Speaker: Luciano Campi (University of Milan, Italy)
Title: Mean Field Games of Singular Control with Finite Fuel
Abstract: We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $\varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework. This talk is based on a joint work with T. De Angelis (Turin University), M. Ghio (SNS, Pisa) and G. Livieri (SNS, Pisa).
Location and Time: Online via Zoom, 4:00pm-5:00pm.
June 23rd, 2021
Speaker: Ulrich Horst (Humboldt University Berlin, Germany)
Title: The Microstructure of Stochastic Volatility Models with Self-Exciting Jump Dynamics
Abstract: We provide a general probabilistic framework within which we establish scaling limits for a class of continuous-time stochastic volatility models with selfexciting jump dynamics. In the scaling limit, the joint dynamics of asset returns and volatility is driven by independednt Gaussian white noise and two independent Poisson random measures that capture the arrival of exogenous shocks and the arrival of selfexcited shocks, respectively. Various well-studied stochastic volatility models with and without selfexciting price/volatility co-jumps are obtained as special cases under different scaling regimes. We analyze the impact of external shocks on the market dynamics, especially their impact on jump cascades and show in a mathematically rigorous manner that many small external shocks may trigger endogenous jump cascades in asset returns and stock price volatility. The talk is based on joint work with Wei Xu.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
June 16th, 2021
Speaker: Peter Caines (McGill University, Canada)
Title: Optimal Execution Problems in Single and Networked Markets: a Mean Field Game Formulation
Abstract: In this work the stock market is modelled as a large population non-cooperative game where each trader has stochastic linear dynamics. We consider the case where there exists one major trader with significant influence on market movements together with a large number of minor traders, each with individually asymptotically negligible effect on the market as the population increases in size. Traders are coupled in their dynamics and in the quadratic cost functions via the market's average trading rate (a component of the system’s mean field). In the first part of the presentation, Mean Field Game theory will be employed to obtain epsilon-Nash equilibria for the market when each agent attempts to (i) maximize its wealth, (ii) track the market's average trading rate, and (iii) avoid large execution prices and large trading accelerations. The generalization to the situation where the agents have only partial (i.e. noisy) observations on their own states and the major agent’s state will be described, including a novel game theoretic belief-of-beliefs feature. In the second part of the talk, we shall present initial results in the application of Graphon Mean Field Game theory to establish epsilon-Nash equilibria for complex networks of markets. Some illustrative simulations will also be presented.
Location and Time: Online via Zoom, 8:30pm-9:30pm.
June 9th, 2021
Speaker: Nizar Touzi (Ecole Polytechnique, France)
Title: Optimal Mutual Holding and Systemic Risk
Abstract: We model the optimal mutual holding problem within a continuum of agent by a mean field game. In the absence of common noise, the problem is solved explicitly by a bang-bang optimal holdig policy. In particular the optimal dynamics are defined by a Mckean-Vlasov SDE with discontinuous diffusion coefficient.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
May 26th, 2021
Speaker: Bruno Bouchard (Université Paris-Dauphine - PSL, France)
Title: Approximate viscosity solutions of path-dependent PDEs and Dupire’s vertical differentiability
Abstract: We introduce a notion of approximate viscosity solution for a class of nonlinear path-dependent PDEs (PPDEs), and provide some existence, uniqueness, comparaison and stability results. In the case of Hamilton-Jacobi type equations, the approximate viscosity solution is naturally related to the associated optimal control problem. We then provide conditions under which the solution admits a Dupire’s vertical derivative and studying its regularity. An application to robust hedging under volatility uncertainty will be discussed.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
May 12th, 2021
Speaker: Huyen Pham (Université de Paris, France)
Title: DeepSet and Their Derivative Networks for Solving Symmetric Problems
Abstract: Machine learning methods represent a breakthrough for solving nonlinear partial differential equations (and control problems in very high dimension, and have been the subject of intense research over the last five years. In this talk, we consider a widespread class of problems that are invariant to permutations of their inputs (state variables or model parameters). This occurs for example in multi asset models for option pricing with exchangeable payoff, for optimal trading portfolio with respect to the market price of covariance risk, or Atlas type models in stochastic portfolio theory Our main application comes actually from mean field control problems and the corresponding PDEs in the Wasserstein space of probability measures Their particle approximations, for which we provide a rate of convergence, lead to symmetric PDEs that are solved by deep learning algorithms based on certain types of neural networks, named DeepSet We illustrate the performance and accuracy of the DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean field systemic risk, and mean variance problem Finally, we show how the combination of DeepSet and DeepOnet, a network architecture recently proposed for learning operators, provides an efficient approximation for a family of optimal trading strategies in terms of market price of covariance risk coefficients.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
April 28th, 2021
Speaker: Daniel Lacker (Columbia University, USA)
Title: A Case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes
Abstract: We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semi-explicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Equilibrium strategies are generically nonlocal, depending on the behavior of all players. Nonetheless, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence. Joint work with Agathe Soret.
Location and Time: Online via Zoom, 8:30pm-9:30pm.
April 14th, 2021
Speaker: Jianfeng Zhang (University of Southern California, USA)
Title: Mean Field Game Master Equation with Non-Separable Hamiltonians and Displacement Monotonicity
Abstract: It is well known that the monotonicity condition is crucial or the global well-posedness of mean field game master equations, and in the literature such conditionis proposed only for separable Hamiltonians. In this talk, we propose a structural condition on non-separable Hamiltonians, in the spirit of the so-called displacement monotonicity condition, and then establish the global well-posedness for the master equation. The talk is based on a joint work with Wilfrid Gangbo, Alpar Meszaros, and Chenchen Mou.
Location and Time: Online via Zoom, 11:00am-12:00pm.