2022 Seminars
Note: We have two talks on Dec 1st.
December 1st, 2022
Speaker: Julio Backhoff (University of Vienna, Austria)
Title: Quantitative Fundamental Theorem of Asset Pricing
Abstract: The aim is to show how to deal with model uncertainty in finance, without imposing the no-arbitrage condition. The idea is to quantify the notion of arbitrage, and obtain a quantitative version of the Fundamental Theorem of Asset Pricing and of the Super-Replication Theorem. As a consequence we can formalize the following statements: in a market that admits "small arbitrage" the "pricing measures'' are such that asset price process is "close to being a martingale'', or equivalently, that hedging strategies need to cover some additional "small costs''. Finally, we study robustness of the amount of arbitrage and existence of respective pricing measures, showing stability of these concepts with respect to an L-infinity version of the adapted Wasserstein distance. Based on joint work with B. Acciaio and G. Pammer.
Date and Time: Thursday, Dec 1st, 2022, 4:00pm-5:00pm.
Location: Online via Zoom
Meeting ID: 948 3563 8896
Passcode: 1201
December 1st, 2022
(Jointly organized with the CUHK Distinguished Lectures in Quantitative Finance)
Speaker: Jakša Cvitanić (California Institute of Technology, USA)
Title: Principal-agent problems in financial markets
Abstract: In this talk, I will present the benchmark continuous time models for contracting between a principal and an agent. Next, I will talk about the extension of the classical models to the case in which the agent controls not only the drift, but also the volatility vector of the output process. Mathematically, this requires results from the theory of second order BSDE's. Then, I will show how to apply this methodology to finding the asset pricing equilibrium and optimal contracts in a market with delegated portfolio management.
Date and Time: Thursday, Dec 1st, 2022, 12:00pm-1:00pm.
Location: Online via Zoom
Meeting ID: 992 2139 1061
Passcode: 343536
November 24th, 2022
Speaker: Frank Riedel (Bielefeld University, Germany)
Title: Efficient Allocations under Ambiguous (Model) Uncertainty
Abstract: We investigate consequences of model uncertainty on ex ante efficient allocations in an exchange economy. The ambiguity we consider is embodied in the model uncertainty perceived by the decision maker: they are unsure what would be the appropriate probability measure to apply to evaluate contingent consumption contingent plans and keep in consideration a set of alternative probabilistic laws. We study the case where the typical consumer in the economy is ambiguity-averse with smooth ambiguity preferences and the set of priors is point identified, i.e., the true law can be recovered empirically from observed events. Differently from the literature, we allow for the case where the aggregate risk is ambiguous and agents are heterogeneously ambiguity averse. Our analysis addresses, in particular, the full range of set-ups where under expected utility the Pareto efficient consumption sharing rule is a linear function of the aggregate endowment. We identify systematic differences ambiguity aversion introduces to optimal sharing arrangements in these environments and also characterize the representative consumer. Furthermore, we investigate the implications for the state-price function, in particular, the effect of heterogeneity in ambiguity aversion.
Date and Time: Thursday, Nov 24th, 2022, 4:00pm-5:00pm.
Location: Online via Zoom
Meeting ID: 928 2539 1869
Passcode: 3485
November 17th, 2022
Speaker: Wenpin Tang (Columbia University, USA)
Title: A prelude to blockchain technology: design and economy
Abstract: A blockchain is a distributed network which functions as a digit ledger and a smart contract allowing the secure transfer of assets without an intermediary. Bitcoin, a P2P electronic cash system, is the first manifestation of the blockchain technology. As the internet is a technology to facilitate the digit flow of information, the blockchain is a technology to facilitate the digital exchange of value. Due to its distributed and secure nature, blockchain technology is believed to be the next generation digital exchange platform with wide applications such as cryptocurrency, healthbank...etc. While the blockchain technology is conceptually powerful, it suffers from two major problems: scalability and security. In this talk, I will first give a "crash course" on the blockchain protocols, e.g. PoW (Proof of Work) and PoS (Proof of Stake). I will then focus on the PoS model, and show how this design may entail different types of risks. I will also discuss a few challenges and research directions in the blockchain designs which are worth further developments. The work is based on joint work with David D. Yao.
Date and Time: Thursday, Nov 17th, 2022, 10:00am-11:00am.
Location: Online via Zoom
Meeting ID: 943 8721 0139
Passcode: 3485
November 10th, 2022
Speaker: Harry Zheng (Imperial College London, United Kingdom)
Title: Duality and Deep Learning for Optimal Consumption with Randomly Terminating Income
Abstract: We establish a rigorous duality theory for a problem of optimal consumption in the presence of an income stream that can terminate randomly at an exponentially distributed time, independent of the asset prices. We thus close a duality gap encountered by Davis and Vellekoop (2009) in a version of this problem in a Black- Scholes market. We then solve the problem numerically, using the primal and dual controls, second order BSDEs, and deep learning, to find the optimal control and tight lower and upper bounds for the value function. (Joint work with Ashley Davey and Michael Monoyios)
Date and Time: Thursday, Nov 10th, 2022, 5:00pm-6:00pm.
Location: Online via Zoom
Meeting ID: 977 0824 2271
Passcode: 1110
November 3rd, 2022
Speaker: Shaolin Ji (Shandong University, China)
Title: Some Recent Progress on Stochastic Maximum Principle with Nonconvex Control Domain
Abstract: Peng (1990) first established a general stochastic maximum principle (SMP) for the classical stochastic control problem with nonconvex control domains. Then some researchers extended the SMP to more general control problems where the state equations are forwardbackward stochastic differential equations (FBSDE). In this presentation, we introduce two recent works. The first issue is that the state equation is a decoupled FBSDE in which the backward equation admits quadratic growth in the argument z. The second issue is that the state equation is a fully coupled FBSDE. For both cases, we study the corresponding stochastic optimal control problems with nonconvex control domains and derive the global SMPs with entirely new terms.
Date and Time: Thursday, Nov 3rd, 2022, 4:00pm-5:00pm.
Location: Online via Zoom
Meeting ID: 995 7854 6852
Passcode: 1103
October 27th, 2022
Speaker: Jean-François Chassagneux (Université Paris Cité, France)
Title: A Dual Approach to Partial Hedging
Abstract: We introduce a class of `weak hedging problem’ which contains as special examples the quantile hedging problem (Föllmer Leukert 1999) and PnL matching problem (introduced in Bouchard & Vu 2012). We show that they can generally be rewritten as a kind of Monge transport problem. Using this observation, we introduce a Kantorovich version of the problem and, in some cases, we are able to prove a dual formulation. This allows us to design numerical methods based on SGD algorithms to compute the weak hedging price. This is a joint work with C. Bénézet (ENSIIE) and M. Yang (Université Paris Cité).
Date and Time: Thursday, Oct 27th, 2022, 3:00pm-4:00pm.
Location: Online via Zoom
Meeting ID: 930 6983 4833
Passcode: 063858
October 20th, 2022
Speaker: Weinan E (Peking University, China)
Title: Deep learning-based Closed-loop Optimal Control Design
Abstract: Progress on designing closed loop optimal control has been hindered by the difficulties associated with the curse of dimensionality. In 2016, Jiequn Han and I developed the first deep learning-based algorithm for solving high dimensional control problems. This opened up a lot of new possibilities including solving high dimensional partial differential equations, high dimensional game theory problems, and of course, high dimensional control problems. In fact, solving high dimensional problems is now among the most active area in scientific computing.
However, it has not been an easy task to turn these kinds of algorithms to practical use. I will review the progress made so far as well as the difficulties in this area. I will discuss applications to the control of robotic arms, as well as to macro-economics.
The work presented is joint work with a lot of people, particularly Jiequn Han, Jihao Long and Yucheng Yang
Date and Time: Thursday, Oct 20th, 2022, 4:00pm-5:00pm.
Location: Online via Zoom
Meeting ID: 945 6531 7300
Passcode: 292730
October 13th, 2022
Speaker: Shige Peng (Shandong University, China)
Title: Robust Brownian Motion in Financial Pricing and Risk Modelling
Abstract: The notion of Brownian motion plays a fundamentally important role in quantitative finance. A well known difficulty in this domain is that it is very hard to be applied in the situation of probability to model uncertainties in this complicated and unpredictable real world. A new notion of robust version of Brownian motion (G-BM) has been introduced. Theoretical investigation and real data implementations showed that this is a robust and powerful tool in financial pricing and risk measuring.
Date and Time: Thursday, Oct 13th, 2022, 4:00pm-5:00pm.
Location: Online via Zoom
Meeting ID: 975 4558 5941
Passcode: 566235
October 6th, 2022
Speaker: Shanjian Tang (Fudan University, China)
Title: Mean Field Games with Mean-Field-Dependent Volatility, and Associated Coupled Nonlocal Quasilinear Forward-Backward Parabolic Equations
Abstract: We consider mean field games with mean-fielddependent volatility, and associated fully coupled nonlocal quasilinear forward-backward PDEs (FBPDEs). We give the global in time existence of classical solutions of the FBPDEs, and the uniqueness under an additional monotonicity condition. A verification theorem is also obtained and the solution of the FBPDEs is used to construct an optimal strategy of the mean field game. Finally, we discuss the linear-quadratic case. This is a joint work with Ziyu Huang, Fudan University.
Date and Time: Thursday, Oct 6th, 2022, 4:00pm-5:00pm.
Location: Online via Zoom
Meeting ID: 915 8016 2767
Passcode: 1006
September 29th, 2022
Speaker: Sebastian Jaimungal (University of Toronto, Canada)
Title: Minimal Kullback-Leibler Divergence for Constrained Lévy-Itô Processes
Abstract: Given an n-dimensional stochastic process X driven by P-Brownian motions and Poisson random measures, we seek the probability measure Q, with minimal relative entropy toP, such that the Q-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterise the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting -- including an algorithm to simulate under the optimal measure -- where an agent seeks to answer the question: what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process? This is joint work with Silvana M. Pesenti and Leandro Sánchez-Betancourt.
Date and Time: Thursday, Sept 29th, 2022, 10:00am-11:00am.
Location: Online via Zoom
Meeting ID: 951 8375 2497
Passcode: 0929
September 14th, 2022
Speaker: Emmanuel Gobet (Ecole Polytechnique, France)
Title: Unbiasing and robustifying implied volatility calibration in a cryptocurrency market with large bid-ask spreads and missing quotes
Abstract: We design a novel calibration procedure that is designed to handle the specific characteristics of options on cryptocurrency markets, namely large bid-ask spreads and the possibility of missing or incoherent prices in the considered data sets. We show that this calibration procedure is significantly more robust and accurate than the standard one based on trade and mid-prices.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
August 11th, 2022
Speaker: Lihu Xu (University of Macau)
Title: A probability approximation framework via Markov process approach
Abstract: We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated Itô’s for- mula and Markov operator. As applications, we study the error bounds of the following approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, approximation of ergodic measure of SDEs driven by stable processes, approximation of ergodic measure of singular SDEs driven by Brownian motion, etc. The tools used in these applications include Duhamel Principle, Malliavin calculus, Zvonkin transform, timechange techniques, etc. This talk is based on the joint work with Peng Chen and Qi-Man Shao.
Location and Time: Online via Zoom, 3:00pm-4:00pm.
July 21st, 2022
Speaker: Yu-Jui Huang (University of Colorado, Boulder)
Title: GANs as Gradient Flows that Converge
Abstract: This paper approaches the unsupervised learning problem by gradient descent in the space of probability density functions. Our main result shows that along the gradient flow induced by a distribution-dependent ordinary differential equation (ODE), the unknown data distribution emerges as the long-time limit of this flow of densities. That is, one can uncover the data distribution by simulating the distribution-dependent ODE. Intriguingly, we find that the simulation of the ODE is equivalent to the training of generative adversarial networks (GANs). The GAN framework between a generator and a discriminator can therefore be viewed alternatively as a cooperative game between a navigator and a calibrator. At the theoretic level, this new perspective simplifies the analysis of GANs and gives new insight into their performance. To construct a solution to the distribution-dependent ODE, we first show that the associated nonlinear Fokker-Planck equation has a unique weak solution, using the Crandall-Liggett theorem in Banach spaces. From this solution to the Fokker-Planck equation, we construct a unique solution to the ODE, relying on Trevisan's superposition principle. The convergence of the induced gradient flow to the data distribution is obtained by analyzing the Fokker-Planck equation. This is joint work with Yuchong Zhang.
Location and Time: Online via Zoom, 11:15am-12:15pm.
July 14th, 2022
Speaker: Mete Soner (Princeton University)
Title: Optimal Stopping in High Dimensions
Abstract: A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several examples related to financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.
Location and Time: Online via Zoom, 8:30pm-9:30pm.
July 7th, 2022
Speaker: Xu Zhang (Sichuan University)
Title: SPDE control: recent progress and open problems
Abstract: In this talk, I will give a short introduction to control theory for stochastic distributed parameter systems (governed by stochastic differential equations in infinite dimensions, typically by stochastic PDEs). I will explain the new phenomena and difficulties in the study of controllability and optimal control problems for these sort of equations. In particular, I will show by some examples that both the formulations of corresponding stochastic control problems and the tools to solve them may differ considerably from their deterministic/finite-dimensional counterparts, and one has to develop new methods, say, the stochastic transposition method introduced in our previous works, to solve some problems in this field.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
June 22nd, 2022
Speaker: Alain Bensoussan (City University of Hong Kong and University of Texas at Dallas)
Title: Stochastic Control and Limited Commitment
Abstract: The theory of investment and growth of firms has been an important source of stochastic control problems. The issue of CEO compensation has been addressed more recently. A seminal paper has been written by H. Ai and R. Li, with a model of CEO compensation under limited commitment. It leads to a new type of stochastic control problem, where a stochastic constraint captures the limited commitment. The authors introduce a Bellman equation, with unusual boundary conditions. Many formal arguments are used in the proof, although the amount of intuition is impressive. The objective of this work is to provide a rigorous and complete theory for this Bellman equation and to solve the corresponding stochastic control problem. Joint work with N. Nguyen and A. Rivera.
Location and Time: Online via Zoom, 11:00am-12:00pm.
June 9th, 2022
Speaker: Martin Larsson (Carnegie Mellon University, USA)
Title: High-Dimensional Open Markets in Stochastic Portfolio Theory
Abstract: Stochastic portfolio theory studies investments in large equity markets. Such investments are frequently confined to an “open market”: a high capitalization investment-grade subset of a much broader equity universe. We develop models for open markets which (i) are consistent with a given invariant distribution of relative market capitalizations, (ii) lead to explicit growth-optimal portfolios, (iii) are robust to the dimensionality and specific characteristics of lowercapitalization stocks outside the investment-grade subset, and (iv) serve as a worst-case model for a robust asymptotic growth maximization problem that incorporates model ambiguity. (Joint work with David Itkin)
Location and Time: Online via Zoom, 8:30pm-9:30pm.
May 26th, 2022
Speaker: Anis Matoussi (Le Mans University, France)
Title: Stochastic Algorithms for Systemic Risk Measures and Application in Insurance
Abstract: In this talk, we discuss the numerical aspects of risk measures for large multivariate systems (multivariate risk sources or heterogeneous actors). Systemic risk measures (SRMs) can be interpreted as the minimum amount of liquidity (cash) that secures a system by allocating capital to each institution before aggregating individual risks. We introduce stochastic algorithms of the Robbins-Monro and Polyak-Ruppert type to approximate SRMs, in particular we show that the estimators used are asymptotically normal. We also present numerical tests to measure the performance of these algorithms (through several examples). This presentation is based on a work in progress with Sarah Kaakai and Achraf Tamtalini (Le Mans University).
Location and Time: Online via Zoom, 4:00pm-5:00pm.
May 19th, 2022
Speaker: Ruodu Wang (University of Waterloo, Canada)
Title: Measuring Diversification via Risk Measures
Abstract: We propose a new notion of diversification indices, called the diversification multipliers (DM). Defined through a parametric family of risk measures, DM satisfy three natural properties, namely non-negativity, location invariance and scale invariance, which are shown to be conflicting for traditional diversification indices based on a single risk measure. We pay special attention to two important classes of risk measures, Value-at-Risk (VaR) and Expected Shortfall (ES). DM based on VaR and ES enjoy many convenient technical properties, and they are easy to compute in portfolio selection problems. The two popular multivariate models of elliptical and regular varying distributions are further analyzed in detail. It turns out that DM can properly distinguish tail heaviness and common shocks, which are neglected by traditional diversification indices. Portfolio optimization with DM is illustrated with financial data and its performance is competitive when contrasted with other diversification methods.
Location and Time: Online via Zoom, 8:30pm-9:30pm.
May 11th, 2022
Speaker: Boualem Djehiche (KTH Royal Institute of Technology, Sweden)
Title: A Particle Approximation of a Class of Mean-field Reflected BSDEs with Jumps
Abstract: I will review a recent result on the propagation of chaos property for weakly interacting nonlinear Snell envelopes which converge to a class of meanfield reflected backward stochastic differential equations (BSDEs) with jumps, where the mean field interaction in terms of the distribution of the Y-component of the solution enters in both the driver and the lower obstacle. The talk is based on a joint work with Roxana Dumitrescu and Jia Zeng.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
May 4th, 2022
Speaker: Zhou Zhou (University of Sydney, Australia)
Title: Equilibrium Concepts for Time-Inconsistent Stopping in Continuous Time
Abstract: A new notion of equilibrium, which we call strong equilibrium, is introduced for time-inconsistent stopping problems in continuous time. Compared to the existing notions of mild equilibrium and weak equilibrium, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous-time Markov chain and the discount function is log sub-additive, we show that an optimal mild equilibrium is always a strong equilibrium, and thus obtain the relations between mild, optimal mild, weak, and strong equilibria. Next, we consider the case when the underlying process is one-dimensional diffusion. We provide necessary and sufficient conditions for the characterization of weak equilibria. The smooth-fit condition is obtained as a by-product. Based on the characterization of weak equilibria, we show that an optimal mild equilibrium is also weak. Then we provide conditions under which a weak equilibrium is strong. We further show that an optimal mild equilibrium is also strong under certain conditions. Finally, we provide several examples including one shows a weak equilibrium may not be strong, and another one shows a strong equilibrium may not be optimal mild.
Location and Time: Online via Zoom, 3:00pm-4:00pm.
April 27th, 2022
Speaker: Junjian Yang (Vienna University of Technology, Austria)
Title: Path-dependent Mean-field Game Optimal Planning
Abstract: In the context of mean-field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P.L. Lions: given a pair of marginal distributions $(\mu_0, \mu_1)$, find a specification of the game problem starting from the initial distribution $\mu_0$, and inducing the target distribution $\mu_1$ at the mean-field game equilibrium. Our main result reduces the path-dependent planning problem into an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals $(\mu_0,\mu_1)$. Some sufficient conditions on $(\mu_0,\mu_1)$ are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along with controlled McKean-Vlasov dynamics.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
April 21st, 2022
Speaker: Ying Jiao (Université Claude Bernard Lyon 1, France)
Title: Alpha-Heston Stochastic Volatility Model
Abstract: We introduce an extension of the Heston model where the instantaneous variance process contains a jump part driven by α-stable processes in order to describe large fluctuations on the market, in particular during the crisis periods. In this framework, we examine the implied volatility surface and its asymptotic behaviors for both asset and variance options. Furthermore, we examine the jump clustering phenomenon and provide a jump cluster decomposition. We show that each cluster process is induced by a first “mother” jump giving birth to a sequence of “child jumps”. We first obtain a closed form for the total number of clusters in a given period. Moreover each cluster process satisfies the same α-CIR evolution of the variance process excluding the long term mean coefficient. Finally, we study the dependence of the number and the duration of clusters as function of the parameter α. This is a joint work with Chunhua Ma, Simone Scotti and Chao Zhou.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
April 13th, 2022
Speaker: Mathieu Rosenbaum (Ecole Polytechnique, France)
Title: Market Making and Incentives Design in the Presence of a Dark Pool: a Deep Reinforcement Learning Approach
Abstract: We consider the issue of a market maker acting at the same time in the lit and dark pools of an exchange. The exchange wishes to establish a suitable make-take fees policy to attract transactions on its venues. We first solve the stochastic control problem of the market maker without the intervention of the exchange. Then we derive the equations defining the optimal contract to be set between the market maker and the exchange. This contract depends on the trading flows generated by the market maker's activity on the two venues. In both cases, we show existence and uniqueness, in the viscosity sense, of the solutions of the Hamilton-Jacobi-Bellman equations associated to the market maker and exchange's problems. We finally design deep reinforcement learning algorithms enabling us to approximate efficiently the optimal controls of the market maker and the optimal incentives to be provided by the exchange.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
April 6th, 2022
Speaker: Renyuan Xu (University of Southern California, USA)
Title: Learning in Linear-Quadratic Framework: From Singleagent to Multi-agent, and to Mean-Field
Abstract: Linear-quadratic (LQ) framework is widely studied in the literature of stochastic control, game theory, and mean-field analysis due to its simple structure, tractable solution, and local approximation power to nonlinear control problems. In this talk, we discuss several theoretical results of the policy gradient (PG) method, a popular reinforcement learning algorithm, for several LQ problems where agents are assumed to have limited information about the stochastic system. In the single-agent setting, we explain how the PG method is guaranteed to learn the global optimal policy. In the multi-agent setting, we show that (a modified) PG method could guide agents to find the Nash equilibrium solution provided there is a certain level of noise in the system. The noise can either come from the underlying dynamics or carefully designed explorations from the agents. Finally, when the number of agents goes to infinity, we propose an exploration scheme with entropy regularization that could help each individual agent to explore the unknown system as well as the behavior of other agents. This talk is based on several projects with Xin Guo (UC Berkeley), Ben Hambly (U of Oxford), Huining Yang (U of Oxford), and Thaleia Zariphopoulou (UT Austin).
Location and Time: Online via Zoom, 3:00pm-4:00pm.
March 24th, 2022
Speaker: Gaoyue Guo (CentraleSupélec, France)
Title: On the Metrics of Wasserstein Type
Abstract: The Wasserstein distance, which arises in optimal transport, is being used more and more in statistics and machine learning. To tackle the curse of dimensionality, the sliced Wasserstein metric and more recently max-sliced Wasserstein metric have attracted abundant attention in data sciences. In this talk, we shall review some applications and provide an analysis on the (strong and topological) equivalence between the metrics of Wasserstein type. This talk is based on the joint work with Erhan Bayraktar.
Location and Time: Online via Zoom, 2:30pm-3:30pm.
March 9th, 2022
Speaker: Asaf Cohen (University of Michigan, USA)
Title: Markovian Equilibria In Ergodic Many-Player Games and Mean-Field Games
Abstract: We consider a symmetric stochastic game with weak interactions between many players. Time is continuous, the number of states is finite, and costs are ergodic. We prove the existence of a unique Nash equilibrium in the game and show that its limiting behavior (as the number of players goes to infinity) is governed by the unique mean-field equilibrium of the corresponding mean-field game. This is joint work with Ethan Zell.
Location and Time: Online via Zoom, 8:30pm-9:30pm.
March 3rd, 2022
Speaker: Hao Xing (Boston University, USA)
Title: Recover Utility of Rational Inattentive Agent and Applications on Robo-advising
Abstract: We consider a rational inattentive agent who acquires costly signal to make decisions. By observing agent’s actions, we formulate an inverse reinforcement learning problem to recover agent’s utility. We propose an efficient numeric algorithm and prove its convergence. The framework is applied to robo-advising problems to recover investors’ utilities by observing their investment strategies. This is a joint work with Zeyu Zhu
Location and Time: Online via Zoom, 8:30pm-9:30pm.
February 23rd, 2022
Speaker: Stefan Ankirchner (University of Jena, Germany)
Title: Diffusion Control Games
Abstract: We consider a symmetric stochastic di fferential game where each player can control the di ffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best alpha of all states receive a fixed prize. Within the mean- field limit version of the game we compute an explicit equilibrium, a threshold strategy that consists in choosing the maximal fluctuation intensity when the state is below a given threshold, and the minimal intensity else. We show that for large n the symmetric n-tuple of the threshold strategy provides an approximate Nash-equilibrium of the n player game. Finally, we compare the approximate equilibrium for large games with the equilibrium of the two player case. The talk is based on joint work with Nabil Kazi-Tani, Julian Wendt and Chao Zhou.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
February 11th, 2022
Speaker: Minyi Huang (Carleton University, Canada)
Title: Linear Quadratic Mean Field Games and Their Asymptotic Solvability
Abstract: We consider linear quadratic (LQ) mean field games (MFGs) and study their asymptotic solvability problems. Roughly, we attempt to answer these questions: When does a sequence of games, with increasing populations, have “well behaved’’ centralized solutions? And how to characterize a necessary and sufficient condition for such nice solution behaviors. We start with a model of homogeneous agents and develop a re-scaling technique for analysis. An important issue in MFGs is the performance of the obtained decentralized strategies in an N-player model, and one usually can obtain an O(N^{-1/2})-Nash equilibrium. By our approach we can improve the estimate from O(N^{-1/2}) to the tightest bound O(1/N). We will further generalize to a major player model and clarify the relation of different solutions existing in the literature. Finally, this asymptotic solvability formulation can be extended to mean field social optimization.
Location and Time: Online via Zoom, 10:00am-11:00am.
January 26th, 2022
Speaker: Johannes Ruf (The London School of Economics and Political Science, UK)
Title: The Growth-Optimal Portfolio in Fund Models
Abstract: This study concerns the estimation of the growth-optimal portfolio. The growth-optimal portfolio, or the numeraire portfolio, maximises the expected logarithmic growth rate and plays a fundamental role in portfolio choice and asset pricing. Efficient estimates of the growth-optimal portfolio under the CAPM and multi-fund models are derived. The accuracy of estimation is greater, the larger the variances of the returns. A measure of investors’ economic loss caused by information loss due to missing data, or unavailable observations is provided. In particular, the economic loss is larger, the larger the investment universe or the factors affecting asset returns. The estimate has a Bayesian interpretation. A shrinkage method targets maximal growth with the least amount of deviation of the growth rates. Joint work with Kostas Kardaras and Hyeng Keun Koo .
Location and Time: Online via Zoom, 4:00pm-5:00pm.
January 19th, 2022
Speaker: Dylan Possamaï (ETH Zurich, Switzerland)
Title: Non-asymptotic Convergence Rates for Mean-field Games: Weak Formulation and McKean–Vlasov BSDEs
Abstract: 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 backward stochastic differential equations, for which we provide a wellposedness 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.
Location and Time: Online via Zoom, 4:00pm-5:00pm.
January 6th, 2022
Speaker: Rainer Buckdahn (Universite de Bretagne Occidentale, France)
Title: Mean-Field BDSDEs and Associated Nonlocal Semi-Linear Backward SPDEs
Abstract: In this talk we investigate mean-field backward doubly stochastic differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also depend on the joint law of the solution process as well as the solution of an associated mean-field forward SDE. We handle a driving coefficient in the backward integral of the BDSDE for which the Lipschitz assumption with respect to the law of the solution is sufficient, without assuming that this Lipschitz constant is small enough. Using the splitting method for mean-field SDEs, we study the unique solutions of our BDSDEs. Under suitable regularity assumptions on the coefficients we investigate the first and the second order derivatives of the solution with respect to x,y and the measure. However, as the parameters run an infinite-dimensional space, unlike Pardoux and Peng, we cannot apply Kolmogorov's continuity criterion to the value function, while in the classical case the value function can be shown to be of class C^{1,2}. Here we have our value function and its derivative are only the L^2-differentiable with respect to x and y, respectively. Moreover, we have to use the mean-field Ito formula. To overcome this problem the characterization of V=(V(t,x,P_{\xi})) as the unique solution of the associated mean-field backward stochastic PDE uses the C_b^{1,2,2}- functions \Psi(t,x,P_{\xi}):=E[V(t,x,P_{\xi})\cdot\eta] for suitable \eta\in L^{\infty}(\mathcal{F};\mathbb{R}). Using a similar idea, we extend the classical mean-field Ito formula to smooth functions of solutions of mean-field BDSDEs. The talk is based on a joint work with Juan Li and Chuanzhi Xing (SDU, Weihai).
Location and Time: Online via Zoom, 4:00pm-5:00pm.