Set-valued Hamilton-Jacobi-Bellman Equations

Building upon the dynamic programming principle for set-valued functions arising from many applications, in this paper we propose a new notion of set-valued PDEs. The key component in the theory is a set-valued It\^{o} formula, characterizing the flows on the surface of the dynamic sets. In the contexts of multivariate control problems, we establish the wellposedness of the set-valued HJB equations, which extends the standard HJB equations in the scalar case to the multivariate case. As an application, a moving scalarization for certain time inconsistent problems is constructed by using the classical solution of the set-valued HJB equation.