Introduction

So far we have studied stochastic differential equations driven by two fundamentally different noise processes, the Wiener process and the Poisson process. The sample paths of the Wiener process are continuous, while those of the Poisson process are discontinuous. The sample paths of a stochastic process x(t) are continuous if its increments, dx, are infinitesimal, meaning that dx → 0 as dt → 0. The Wiener and Poisson processes have two properties in common. The first is that the probability densities for their respective increments do not change with time, and the second is that their increments at any given time are independent of their increments at all other times. The increments of both processes are thus mutually independent and identically distributed, or i.i.d. for short. In Section 3.3, we discussed why natural noise processes that approximate continuous i.i.d. processes are usually Gaussian, and that this is the result of the central limit theorem. In this chapter we consider all possible i.i.d. noise processes. These are the Levy processes, and include not only the Gaussian and Poisson (jump) processes that we have studied so far, but processes with continuous sample paths that do not obey the central limit theorem.

There are three conditions that define the class of Levy processes. As mentioned above, the infinitesimal increments, dL, for a given Levy process, L(t), are all mutually independent and all have the same probability density.