  Using a Markov duality satisfied by ASEP on the integer line, we deduce a
similar Markov duality for half-line open ASEP and open ASEP on a segment. This
leads to closed systems of ODEs characterizing observables of the models. In
the half-line case, we solve the system of ODEs using Bethe ansatz and prove an
integral formula for $q$-moments of the current at $n$ distinct spatial
locations. We then use this formula to confirm predictions for the moments of
the multiplicative noise stochastic heat equation on $\mathbb R_{>0}$ with
Robin type boundary condition and we obtain new formulas in the case of a
Dirichlet boundary condition.
