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We present a simple construction method for Feller processes and a frameworkfor the generation of sample paths of Feller processes. The construction isbased on state space dependent mixing of L\'evy processes.  Brownian Motion is one of the most frequently used continuous time Markovprocesses in applications. In recent years also L\'evy processes, of whichBrownian Motion is a special case, have become increasingly popular.  L\'evy processes are spatially homogeneous, but empirical data often suggestthe use of spatially inhomogeneous processes. Thus it seems necessary to go tothe next level of generalization: Feller processes. These include L\'evyprocesses and in particular Brownian motion as special cases but allow spatialinhomogeneities.  Many properties of Feller processes are known, but proving the very existenceis, in general, very technical. Moreover, an applicable framework for thegeneration of sample paths of a Feller process was missing. We explain, withpractitioners in mind, how to overcome both of these obstacles. In particularour simulation technique allows to apply Monte Carlo methods to Fellerprocesses.

Feller Processes: The Next Generation in Modeling. Brownian Motion, L\'evy Processes and Beyond