the Levy-Ito decomposition
/ lev-EE ee-TOH /
The Lévy-Khintchine formula tells you the law of a Lévy process; the Lévy-Itô decomposition tells you its paths. It is the pathwise realization of the triplet (b, sigma^2, nu): it builds an arbitrary Lévy process explicitly as a sum of a deterministic drift, an independent Brownian motion, and an independent jump part assembled from a Poisson random measure of jumps.
The theorem states that any Lévy process X_t can be written, almost surely, as X_t = b t + sigma B_t + integral over (|x| >= 1) of x N(t, dx) + integral over (0 < |x| < 1) of x (N(t, dx) - t nu(dx)). Here B_t is a standard Brownian motion, N(t, dx) is the Poisson random measure counting the jumps of X up to time t with intensity dt nu(dx), the third term is the (finite) sum of large jumps, and the fourth term is the compensated integral of small jumps, which converges in L^2 as a martingale precisely because integral over (|x|<1) of x^2 nu(dx) < infinity. The four pieces are mutually independent, and the continuous part (b t + sigma B_t) is independent of the jump part. Large jumps must be added raw (their compensator could diverge if nu has heavy tails), while small jumps must be compensated (their raw sum could diverge if they are too numerous near zero) — this is the operational meaning of the small-versus-large split.
This is the structural theorem of the field: it justifies thinking of a Lévy process as continuous diffusion plus jumps, it is the route to the Itô formula for jump processes, and it underlies simulation (simulate Brownian motion, simulate large jumps as a compound Poisson process, approximate small jumps). The subtlety to respect: the small-jump term is generally NOT a sum of its jumps in the ordinary sense — when integral of |x| nu(dx) diverges near zero, only the COMPENSATED sum converges, and dropping the compensator gives a divergent, meaningless expression.
A gamma process has Lévy measure nu(dx) = c x^(-1) e^(-x) dx on x > 0; near zero this integrates x^(-1) so the jumps are not summable, yet x nu(dx) ~ c e^(-x) dx integrates, making it finite variation. Its Lévy-Itô form has sigma = 0, no compensation needed, and the process is the increasing limit of its (countably infinitely many) positive jumps.
Drift + Brownian + large jumps + compensated small jumps, made path-explicit.
The split point |x| = 1 is a convention; moving it changes only the drift b. The compensation of small jumps is mandatory exactly when nu is not integrable near zero — otherwise you may write the jump part as an ordinary sum and absorb its mean into b.