  We consider the problem of nonlinear stochastic optimal control. This problem
is thought to be fundamentally intractable owing to Bellman's ``curse of
dimensionality". We present a result that shows that repeatedly solving an
open-loop deterministic problem from the current state with progressively
shorter horizons, similar to Model Predictive Control (MPC), results in a
feedback policy that is $O(\epsilon^4)$ near to the true global stochastic
optimal policy, \nxx{where $\epsilon$ is a perturbation parameter modulating
the noise.} We show that the optimal deterministic feedback problem has a
perturbation structure in that higher-order terms of the feedback law do not
affect lower-order terms, and that this structure is lost in the optimal
stochastic feedback problem. Consequently, solving the Stochastic Dynamic
Programming problem is highly susceptible to noise, even when tractable, and in
practice, the MPC-type feedback law offers superior performance even for
stochastic systems.
