This paper provides new insights into the solution of optimal stochastic control problems by means of
a system of partial differential equations, which characterize directly the optimal control. This new
system is obtained by the application of the stochastic maximum principle at every initial condition,
assuming that the optimal controls are smooth enough. The type of problems considered are those
where the diffusion coefficient is independent of the control variables, which are supposed to be
interior to the control region. The results obtained are applied to the study of the classical
consumption–savings model
