Abstract

An optimal control problem with (possibly) unbounded terminal cost is considered in 2(Wd), the space of Borel probability measures with finite second moment. We consider a suitable weak topology rendering 2(Wd) 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 (HJB) equation. Additionally, if the terminal cost is bounded and continuous, we show that the value function is the unique viscosity solution of the HJB equation.