Stochastic control problems (SCPs) provide a versatile framework for studying problems in finance and engineering, among other domains. Typically, the dynamics of the controlled process X are described by a Stochastic Differential Equation (SDE) with an initial condition, and the optimal control process is determined through the maximization of a reward functional. A relatively forsaken subclass of SPCs is those involving target constraints. These constraints can take on finite-dimensional forms, such as w(X_T, Y_T)=0 for a function w and a target Y_T. Alternatively, they can be infinite-dimensional, such as X_T~\mufin, where ~ denotes equality in distribution and \mufin denotes a target distribution.
In this talk, we elucidate how different tools from the analysis of SCPs with target constraints unlock the study of two problems: Stackelberg games with private information and the mean-field Schrödinger problem. Our approach for addressing the first application relies on the dynamic programming principle and the analysis of backward SDEs to find a reformulation of the problem that is amenable to a verification theorem in terms of a well-specified system of Hamilton–Jacobi–Bellman equations. To handle the inherent infinite-dimensional nature of the target constraint in the second problem, we leverage the stochastic maximum principle and the analysis of McKean–Vlasov forward-backward SDEs.
Bio: Camilo Hernandez is a Presidential Postdoc in the Department of Operations Research and Financial Engineering. He has published his work on economic behavioral features such as time inconsistency and its implications to contract theory and finance, and he has written about propagation of chaos in systems of interacting particles in the context of the so-called Schrödinger problem. Before coming to Princeton, Camilo was a Chapman Fellow in Mathematics at Imperial College London. He holds a Ph.D. in Operations Research from Columbia University, a master's degree in Mathematics from the Universidad de los Andes, in Bogotá, Colombia, as well as Bachelor's of Science degrees in both Economics and Mathematics from there.