Seminar by Camilo Hernández

Target Constraints in Stochastic Control Problems and Their Applications

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~𝜇, where ~ denotes equality in distribution and 𝜇 denotes a target distribution.

In this talk, motivated by the study of contract theory models with constrained contracts and entropy-regularized optimal transport, we elucidate how different tools from control theory unlock the analysis of two problems: Stackelberg games and the mean-field Schrödinger problem. Our approach for addressing the first problem 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 Hamilton–Jacobi–Bellman equation with an a priori unknown domain. 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.