  By extending \cite{bensoussan2015control}, we implement the proposal of Lions
\cite{lions14} on studying mean field games and their master equations via
certain control problems on the Hilbert space of square integrable random
variables. In \cite{bensoussan2015control}, the Hilbert space could be quite
general in the face of the "deterministic control problem" due to the absence
of additional randomness; while the special case of $L^2$ space of square
integrable random variables was brought in at the interpretation stage. The
effectiveness of the approach was demonstrated by deriving Bellman equations
and the first order master equations through control theory of dynamical
systems valued in the Hilbert space. In our present problem for second order
master equations, it connects with a stochastic control problem over the space
of random variables, and it possesses an additional randomness generated by the
Wiener process which cannot be detached from the randomness caused by the
elements in the Hilbert space. Nevertheless, we demonstrate how to tackle this
difficulty, while preserving most of the efficiency of the approach suggested
by Lions \cite{lions14}.
