Session: Dynamic Optimization: Deterministic and Stochastic Continuous-time Models I
Chair: Cristopher Hermosilla
Cluster: Multi-agent Optimization and Games
Talk 1: A Minimality Property of the Value Function in Optimal Control over the Wasserstein Space
Speaker: Cristopher Hermosilla
Abstract: In this talk we study an optimal control problem with (possibly) unbounded terminal cost in the space of Borel probability measures with finite second moment. We consider the metric geometry associated with the Wasserstein distance, and a suitable weak topology rendering this space locally compact. In this setting, we show that the value function of a control problem is the minimal viscosity supersolution of an appropriate Hamilton-Jacobi-Bellman equation. Additionally, if the terminal cost is bounded and continuous, we show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.
Talk 2: Principal-Multiagents problem in continuous-time
Speaker: Nicolás Hernández
Abstract: We study a general contracting problem between the principal and a finite set of competitive agents, who perform equivalent changes of measure by controlling the drift of the output process and the compensator of its associated jump measure. In this setting, we generalize the dynamic programming approach developed by Cvitanić, Possamaï, and Touzi (2017) and we also relax their assumptions. We prove that the problem of the principal can be reformulated as a standard stochastic control problem in which she controls the continuation utility (or certainty equivalent) processes of the agents. Our assumptions and conditions on the admissible contracts are minimal to make our approach work. We also present a smoothness result for the value function of a risk–neutral principal when the agents have exponential utility functions. This leads, under some additional assumptions, to the existence of an optimal contract.
Talk 3: Unbounded viscosity solutions of Hamilton-Jacobi equations in the 2-Wasserstein space
Speaker: Othmane Jerhaoui
Abstract: In this talk, we study unbounded viscosity solutions of Hamilton-Jacobi equations in the 2-Wasserstein space over the Euclidean space. The notion of viscosity is defined by taking test functions that are locally Lipschitz and can be respresented as a difference of two geodesically semiconvex functions. First, We establish a comparison result for a general Hamiltonian sat- isfying mild hypotheses. Then, we will discuss well-posedness of a class of Hamilton-Jacobi equations with a Hamiltonian arising from classical mechanics.
