In this talk, I will introduce the hydrodynamic Euler–Alignment model, focusing on the pressureless case coupled with nonlocal interaction forces, and discuss its large-time dynamics—namely, the emergence of flocking and the characterization of its asymptotic behavior.

New flocking estimates will be presented, showing how the confining effect of nonlocal interaction can, in certain regimes, replace the role of velocity alignment.

The quantitative analysis of the asymptotic behavior will also be discussed. Overall, the convergence rate depends only on the local behavior of the communication weight: bounded kernels lead to exponential decay, while weakly singular ones yield algebraic rates. This reveals a sharp transition in decay rates driven solely by the local singularity of the communication kernel, a regime that had remained largely unexplored.

This talk is based on joint work with José Carrillo (University of Oxford), Young-Pil Choi (Yonsei University), and Oliver Tse (Eindhoven University of Technology).

We study optimal stopping for diffusion processes with unknown model primitives within the continuous-time reinforcement learning (RL) framework developed by Wang et al. (2020), and present applications to option pricing and portfolio choice. By penalizing the corresponding variational inequality formulation, we transform the stopping problem into a stochastic optimal control problem with two actions. We then randomize controls into Bernoulli distributions and add an entropy regularizer to encourage exploration. We derive a semi-analytical optimal Bernoulli distribution, based on which we devise RL algorithms using the martingale approach established in Jia and Zhou (2022a). We establish a policy improvement theorem and prove the fast convergence of the resulting policy iterations. We demonstrate the effectiveness of the algorithms in pricing finite-horizon American put options, solving Merton’s problem with transaction costs, and scaling to high-dimensional optimal stopping problems. In particular, we show that both the offline and online algorithms achieve high accuracy in learning the value functions and characterizing the associated free boundaries.

 

Joint work with Min Dai, Yu Sun and Zuo Quan Xu, and forthcoming in Management Science.