  We consider a stochastic transportation problem between two prescribed
probability distributions (a source and a target) over processes with general
drift dependence and with free end times. First, and in order to establish a
dual principle, we associate two equivalent formulations of the primal problem
in order to guarantee its convexity and lower semi-continuity with respect to
the source and target distributions. We exhibit an equivalent Eulerian
formulation, whose dual variational principle is given by
Hamilton-Jacobi-Bellman type variational inequalities. In the case where the
dependence on the drift is bounded, regularity results on the minimizers of the
Eulerian problem then enable us to prove attainment in the corresponding dual
problem. We also address attainment when the drift component of the cost
defining Lagrangian $L$ is superlinear $L \approx |u|^p$ with $1<p<2$, in which
case the setting is reminiscent of our approach -- in a previous work -- on
deterministic controlled transport problems with free end time. We finally
address criteria under which the optimal drift and stopping time are unique,
namely strict convexity in the drift component and monotonicity in time of the
Lagrangian.
