Continuous-time dynamic-programming PDE for the value function V(t,x) of a stochastic control problem: 0 = dfracpartial Vpartial t + sup_u!left mathcalL^u V + f(x,u) right, where mathcalL^u is the controlled infinitesimal generator and f the running reward. Underlies Merton's optimal consumption/investment problem (with closed-form CRRA solution), optimal execution (Almgren-Chriss), and reinforcement-learning-style hedging. Verification theorem links smooth solutions of HJB to optimal controls; viscosity solutions handle the non-smooth case.