  We analyze an optimal stopping problem with a constraint on the expected
cost. When the reward function and cost function are Lipschitz continuous in
state variable, we show that the value of such an optimal stopping problem is a
continuous function in current state and in budget level. Then we derive a
dynamic programming principle (DPP) for the value function in which the
conditional expected cost acts as an additional state process. As the optimal
stopping problem with expectation constraint can be transformed to a stochastic
optimization problem with supermartingale controls, we explore a second DPP of
the value function and thus resolve an open question recently raised in [S.
Ankirchner, M. Klein, and T. Kruse, A verification theorem for optimal stopping
problems with expectation constraints, Appl. Math. Optim., 2017, pp. 1-33].
Based on these two DPPs, we characterize the value function as a viscosity
solution to the related fully non-linear parabolic Hamilton-Jacobi-Bellman
equation.
