Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes I

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.