Derivation — The Lévy-Itô Decomposition and the Lévy Triple

The Lévy-Itô decomposition says every Lévy process — any process with stationary, independent increments — can be decomposed into exactly three independent pieces: a deterministic drift, a Brownian component, and two jump components (small jumps compensated to remove their mean, large jumps uncompensated). The Lévy triple $(\gamma, \sigma^2, \nu)$ that characterises these pieces is the fingerprint of the process: give the triple, and you have completely described the law of the process at every time horizon.

This note derives the decomposition and the Lévy-Khintchine characteristic function formula. Merton and Kou are special cases — their Lévy triples have $\nu(dx) = \lambda f_Y(x)\,dx$ for specific $f_Y$.