Feller processes of normal inverse Gaussian type

We consider the construction of normal inverse Gaussian (NIG) (and some related) Levy processes from the probabilistic viewpoint and from that of the theory of pseudo-differential operators; we then introduce and analyse natural generalizations of these constructions. The resulting Feller processes are somewhat similar to the NIG Levy process but may, for instance, possess mean-reverting features. Possible applications to financial mathematics are discussed, and approximations to solutions of corresponding generalizations of the Black-Scholes equation are derived.