Given a stochastic state process
(
X
t
)
t
(X_t)_t
and a real-valued submartingale cost process
(
S
t
)
t
(S_t)_t
, we characterize optimal stopping times
τ
\tau
that minimize the expectation of
S
τ
S_\tau
while realizing given initial and target distributions
μ
\mu
and
ν
\nu
, i.e.,
X
0
∼
μ
X_0\sim \mu
and
X
τ
∼
ν
X_\tau \sim \nu
. A dual optimization problem is considered and shown to be attained under suitable conditions. The optimal solution of the dual problem then provides a contact set, which characterizes the location where optimal stopping can occur. The optimal stopping time is uniquely determined as the first hitting time of this contact set provided we assume a natural structural assumption on the pair
(
X
t
,
S
t
)
t
(X_t, S_t)_t
, which generalizes the twist condition on the cost in optimal transport theory. This paper extends the Brownian motion settings studied in Ghoussoub, Kim, and Palmer [Calc. Var. Partial Differential Equations 58 (2019), Paper No. 113, 31] and Ghoussoub, Kim, and Palmer [A solution to the Monge transport problem for Brownian martingales, 2019] and deals with more general costs.