| Abstract: |
| 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, 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. |
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