Lévy-Khintchine Representation for Lévy Processes

Students often conflate the Lévy measure

\nu(dx)

with a probability distribution. It is not; it is a measure that can be infinite near the origin. The compensation term

- i \langle u, x \rangle \mathbb{1}_{\{|x|<1\}}

is essential to ensure the integral converges, essentially 'centering' the small jumps.

Institutional Reference: Advanced Stochastic Processes