Construction of Lévy Processes

Abstract

Our starting point is now the Lévy–Khintchine formula for the characteristic exponent ψ of a Lévy process

$$\psi(\xi)\;=\;- il\cdot\xi\;+\;\frac{1}{2}\xi\cdot Q\xi\;+\;\int_{y\neq 0}[1-e^{iy\cdot\xi}\;+\;i\xi\cdot y1\!\!\!\!1_{(0,1)}(|y|)]v(dy)$$

(7.1)

where (l,Q, ν) is a Lévy triplet in the sense of Definition 6.10; a proof of (7.1) is contained in Theorem 6.8, but the exposition below is independent of this proof, see however Remark 7.7 at the end of this chapter.