Abstracts of Talks 

Mete Soner

Title: Synchronization games, Markov Perfect Equilibria and time discretization

Abstract: Building on Winfree's work, the Kuramoto model (1975) has become the corner stone of mathematical models of collective synchronization, and has received attention in all natural sciences, engineering, and mathematics. While the classical model postulates the dynamics of each oscillator in the form of a system of nonlinear ordinary differential equations, one may design games mimicking similar behavior: the system is unsynchronized when the coupling is not sufficiently strong, fascinatingly, they exhibit an abrupt transition to synchronization above a critical value of the interaction parameter.  In this talk, I describe the general problem and then specialize to simple two-state model with a remarkably rich structure.  Through this example, I introduce the discrete time, finite player games with a finite state-space, and discuss the corresponding  Markov Perfect Equilibria (MPE).  In particular, MPE are always unique in continuous time and iterative schemes converge exponentially.  Interestingly, in discrete time these properties hold only when the time step is sufficiently small.

This talk is based on joint works with Rene Carmona, Felix Hoefer, and Qinxin Yan of Princeton, Quentin Cormier of INRIA, Mathieu Lauriere of NYU-Shanghai, and Atilla Yilmaz of Temple.

Xin Zhang

Title: Exciting games and Monge-Ampère equations

Abstract: In this talk, we consider a competition between d+1 players, and aim to identify the “most exciting game” of this kind. This is translated, mathematically, into a stochastic optimization problem over  martingales that live on the d-dimensional sub-probability simplex  and terminate on the vertices of the simplex, with a cost function related to a scaling limit of Shannon entropies. We uncover a surprising connection between this problem and the seemingly unrelated field of Monge-Ampère equations, and identify the optimal martingale via a detailed analysis of boundary asymptotics of a Monge-Ampère equation.

Jiamin Jian

Title: Turnpike properties in linear quadratic Gaussian N-player differential games

Abstract: We consider the long-time behavior of equilibrium strategies and state trajectories in a linear quadratic N-player game with Gaussian initial data. By analyzing convergence toward the corresponding ergodic game, we establish exponential convergence estimates between the solutions of the finite-horizon Riccati system and the associated algebraic Riccati system arising in the ergodic setting. Building on these results, we prove the convergence of the time-averaged value function and derive a turnpike property for the equilibrium pairs of each player. Importantly, our approach avoids reliance on the mean field game limiting model, allowing for a fully uniform analysis with respect to the number of players N. As a result, we further establish a uniform turnpike property for the equilibrium pairs between the finite-horizon and ergodic games with N players.

Xiaofei Shi

Title: Illiquidity, Interest Rates, and Asset Prices

Abstract: In this talk we present how the interest rates and the price dynamics of a risky asset depend on the market illiquidity. In particular, with the presence of even small trading costs, the interest rates determined in equilibrium will become stochastic. An equilibrium is achieved through a system of coupled forward-backward SDEs, whose solution turns out to be amenable to an asymptotic analysis for the practically relevant regime of large liquidity. These tractable approximation formulas match with stylized facts observed in reality. (Joint work in progress with Paolo Guasoni and Johannes Muhle-Karbe.)

Fang Rui Lim

Title: Causal transports on path space and related transport problems

Abstract: Causal optimal transport and its induced adapted Wasserstein distance have recently been gaining more attention as a viable alternative to the usual distances between laws of stochastic processes. This is, in part, due to its applicability to dynamic optimization problems arising from mathematical finance and stochastic analysis, including optimal stopping and mean variance hedging. 

In this talk, we characterize the (bi-)causal transports between the laws of many familiar stochastic processes, such as Brownian motion and the solutions to various stochastic equations. For example, we show that bi-causal maps between the laws of Brownian motion are, in fact, stochastic integrals of rotation-valued integrands. This representation allows us to explicitly compute the adapted Wasserstein distance between the laws of mean-square continuous Gaussian processes. We also solve the optimal (bi-)causal transport problem between the laws of solutions to scalar stochastic equations under various monotonicity conditions.

This is joint work with Professor Rama Cont and Yifan Jiang.

Daniel Lacker

Title: Delocalization of bias in high-dimensional Langevin Monte Carlo

Abstract: The error of the Euler discretization for a high-dimensional stochastic differential equation (SDE) is known to scale with the dimension. In particular, the same is true for the unadjusted Langevin algorithm, which uses the Euler scheme to generate approximate samples from the invariant measure of the SDE. A recent paper of Chen, Cheng, Niles-Weed, and Weare identifies an intriguing new delocalization phenomenon: For certain classes of dynamics, the approximation error for low-dimensional marginals scales only with the lower dimension, not the full dimension. This talk will explain this new phenomenon and our recent results, which strengthens those of Chen et al. in several directions. The proofs are based on a hierarchical analysis of the marginal relative entropies, inspired by our recent work on propagation of chaos. Joint work with Fuzhong Zhou.

Sebastian Jaimungal

Title: Kullback-Leibler Barycentre of Stochastic Processes

Abstract: We consider the problem where an agent aims to combine the views and insights of different experts' models. Specifically, each expert proposes a diffusion process over a finite time horizon. The agent then combines the experts' models by minimising the weighted Kullback--Leibler divergence to each of the experts' models. We show existence and uniqueness of the barycentre model and prove an explicit representation of the Radon--Nikodym derivative relative to the average drift model. We further allow the agent to include their own constraints, resulting in an optimal model that can be seen as a distortion of the experts' barycentre model to incorporate the agent's constraints. We propose two deep learning algorithms to approximate the optimal drift of the combined model, allowing for efficient simulations. The first algorithm aims at learning the optimal drift by matching the change of measure, whereas the second algorithm leverages the notion of elicitability to directly estimate the value function. The paper concludes with an extended application to combine implied volatility smile models that were estimated on different datasets.

[ based on joint work with Silvana Pesenti ]

Max Reppen

Title: Before the Storm: Firm Policies and Varying Recession Risk

Abstract: Recession risk fluctuates substantially “before the storm,” yet little is known about how firms of different sizes adapt their policies accordingly. We embed time-varying recession risk into a model of liquidity management and investment, allowing firms to time their precautionary policies. Estimation reveals that small firms are less sensitive than large firms in their issuance, payout, and investment strategies to variations in recession risk. This is because small firms proactively build larger cash reserves relative to their current cash flow volatility during periods of low recession risk, anticipating that aggressive investment will gradually drain liquidity, grow cash flow volatility and raise the risk of liquidation if a recession occurs. In contrast, large firms rely on robust cash flows to replenish liquidity during periods of low recession risk, deferring other precautionary actions until recession risk intensifies. These results have important implications for estimating the impact of recessions and macroprudential policy.

Silvana Pesenti

Title: Marginal fairness: Fair decision-making under risk measures

Abstract: Making fair or discrimination-free decisions is of concern in many applications such as finance, insurance, and machine learning tasks. This talk is on marginal fairness, a new individual fairness notion for equitable decision-making in the presence of protected attributes such as gender, race, and religion. The criterion ensures that decisions based on distortion risk measures are insensitive to distributional perturbations in protected attributes, regardless of whether these attributes are continuous, discrete, categorical, univariate, or multivariate. To operationalize this notion and reflect real-world regulatory environments (such as the EU gender-neutral pricing regulation),  we model business decision-making in highly regulated industries (such as insurance and finance) as a two-step process: (i) a predictive modeling stage, in which a prediction function for the target variable (e.g., credit risk loss) is estimated based on both protected and non-protected covariates; and (ii) a decision-making stage, in which a distortion risk measure is applied to the target variable, conditional only on non-protected covariates, to determine the decision. In this second step, we modify the risk measure such that the decision becomes insensitive to the protected attribute, thus enforcing fairness to ensure equitable outcomes under risk-sensitive, regulatory constraints. Furthermore, by utilizing the concept of cascade sensitivity, we extend the marginal fairness framework to capture how dependencies between covariates propagate the influence of protected attributes through the modeling pipeline. An empirical implementation using an auto insurance dataset demonstrate how the framework can be applied in practice.

This is joint work with Fei Huang, UNSW.

Yuqiong Wang

Title: On hypoellipticity of degenerate operators in testing and detection problems

Abstract: We study a family of stopping problems that arise in Bayesian sequential testing and quickest detection. One observes a $k-$dimensional Brownian motion whose drift is unknown and determined by a Markov process $\{\theta_t: t\geq 0\}$ that has $n+1$ different states, and the payoff function depends on the true state. We assume that $k<n$, i.e., the state dimension exceeds the number of Brownian drivers, which makes the problem degenerate. In the posterior likelihood coordinate, we give conditions under which the operator is hypoelliptic in the interior, identify cases where it must fail, and discuss their consequences. We present a few illustrative examples that fall into this category. This is joint work with Erhan Bayraktar.

Valentin Tissot-Daguette

Title: Relaxed MOT and Mocking Martingales

Abstract: Martingale Optimal Transport (MOT) offers a powerful, robust framework to price and hedge illiquid derivatives. In the primal formulation, admissible models are required to exactly match the market’s implied volatility (IV) skews, thus imposing hard constraints on the marginal distributions. In practice, it is sufficient that the model’s IVs of vanilla options fall within the observed bid-ask range. This translates mathematically into the bid, model, and ask marginals being in convex order, leading to a relaxed version of MOT. By enlarging the set of admissible martingale couplings, Relaxed MOT generates wider, yet more realistic price bounds on illiquid instruments as shown in the practical examples. We also introduce a weaker notion of mimicking martingales, coined mocking martingales, and extend Kellerer’s theorem to characterize their existence. 

 Joint work with Shunan Sheng, Marcel Nutz (Columbia), and Bryan Liang (Bloomberg). 

Ali Kara

Title: Linear Function Approximations in Partially Observed Reinforcement Learning

Abstract: Control of partially observed Markov decision process (POMDPs) suffers from both the curse of history and the curse of dimensionality, since an optimal control generally depends on the entire history of observables.  Finite-memory control strategies can be near-optimal under suitable filter stability conditions, but they still face dimensionality challenges even when the observation space is finite and small.

I will talk about the standard approach of linear function approximation for learning in partially observed control problems. We will see that in this setting; the learning algorithms track the composition of an approximate Bellman operator with a projection operator. For policy evaluation, this composition can be shown to be contractive in L_2 norm of the stationary distribution of the finite-memory variables. Consequently, the learning method converges to an estimate that is sufficiently close to the true value of the policy under appropriate conditions.

However, for optimal value estimation, the composition operator is generally not contractive due to the mismatch between the exploration policy and the greedy policy implicit in the Bellman operator. As a result, the learning algorithms, in general, may be unstable except in certain special cases. These special cases include: (i) when the basis functions are orthogonal, (ii) when the output of Bellman operator remains in the span of the basis functions.

Ioannis Karatzas

Title: Degenerate Competing Three-Particle Systems

Abstract: We study the process of gaps in a degenerate system of three particles interacting through their ranks, and obtain the Laplace transform of its invariant measure as well as an explicit expression for the corresponding invariant density. We start from the basic adjoint relationship characterizing the invariant measure, then apply a combination of two approaches: the invariance methodology of W. Tutte, thanks to which we compute the Laplace transform in closed form; and a recursive compensation approach which leads to the density of the invariant measure in the form of an infinite sum of exponentials. As in the case of Brownian motion with reflection or killing at the endpoints of an interval, certain Jacobi theta-type functions play a crucial role in our computations. (Joint work with Sandro Franceschi, Tomoyuki Ichiba and Killian Rachel.)