  We connect boundary conditions for one-sided pseudo-differential operators
with the generators of modified one-sided L\'evy processes. On one hand this
allows modellers to use appropriate boundary conditions with confidence when
restricting the modelling domain. On the other hand it allows for numerical
techniques based on differential equation solvers to obtain fast approximations
of densities or other statistical properties of restricted one-sided L\'evy
processes encountered, for example, in finance. In particular we identify a new
nonlocal mass conserving boundary condition by showing it corresponds to
fast-forwarding, i.e. removing the time the process spends outside the domain.
We treat all combinations of killing, reflecting and fast-forwarding boundary
conditions.
  In Part I we show wellposedness of the backward and forward Cauchy problems
with a one-sided pseudo-differential operator with boundary conditions as
generator. We do so by showing convergence of Feller semigroups based on grid
point approximations of the modified L\'evy process.
  In Part II we show that the limiting Feller semigroup is indeed the semigroup
associated with the modified L\'evy process by showing continuity of the
modifications with respect to the Skorokhod topology.
