  We study a stochastic control/stopping problem with a series of
inequality-type and equality-type expectation constraints in a general
non-Markovian framework. We demonstrate that the stochastic control/stopping
problem with expectation constraints (CSEC) is independent of a specific
probability setting and is equivalent to the constrained stochastic
control/stopping problem in weak formulation (an optimization over joint laws
of Brownian motion, state dynamics, diffusion controls and stopping rules on an
enlarged canonical space). Using a martingale-problem formulation of controlled
SDEs in spirit of \cite{Stroock_Varadhan}, we characterize the probability
classes in weak formulation by countably many actions of canonical processes,
and thus obtain the upper semi-analyticity of the CSEC value function.
  Then we employ a measurable selection argument to establish a dynamic
programming principle (DPP) in weak formulation for the CSEC value function, in
which the conditional expected costs act as additional states for constraint
levels at the intermediate horizon. This article extends the results of
\cite{Elk_Tan_2013b} to the expectation-constraint case. We extend our previous
work \cite{OSEC_stopping} to the more complicated setting where the diffusion
is controlled. Compared to that paper the topological properties of
diffusion-control spaces and the corresponding measurability are more
technically involved which complicate the arguments especially for the
measurable selection for the super-solution side of DPP in the weak
formulation.
