The policy of an optimal control problem for
nonlinear stochastic systems can be characterized by a second-
order partial differential equation for which solutions are not
readily available. In this paper we provide a systematic method
for obtaining approximate solutions for the infinite-horizon
optimal control problem in the stochastic framework. The
method is demonstrated on an illustrative numerical example
in which the control effort is not weighted, showing that the
technique is able to deal with one of the most striking features
of stochastic optimal control.