Approximation of Discontinuous Processes by Continuous Processes

There is a close relationship between methods of characterizing stochastic processes and methods for justifying approximations. In this paper we will consider two methods of characterizing processes, (1) as solutions of stochastic equations involving multidimensional random time transformations, and (2) (for Markov processes) by specifying their generators, and we will consider examples of how these methods lead naturally to approximation theorems.