The Lévy-Khintchine Representation

Brownian motion, the Poisson process, and Merton's jump-diffusion model all look like unrelated objects the first time you meet them, yet they're all special cases of one family: Lévy processes — any process with independent, stationary increments and no gaps in probability over small time steps. The Lévy-Khintchine representation is the theorem that says every member of this entire family, however exotic its jump behavior, can be broken down into exactly three ingredients, expressed through its characteristic function:

$$\log E\left[e^{iuX_t}\right] = t \left( ib u - \tfrac{1}{2}\sigma^2 u^2 + \int \left(e^{iux}-1-iux\,\mathbb{1}_{|x|<1}\right)\nu(dx) \right).$$

In plain English, the three ingredients are: a deterministic drift, $b$; a continuous, Gaussian diffusion component with variance rate $\sigma^2$ (ordinary Brownian motion); and a jump component governed entirely by the Lévy measure $\nu$, which specifies the rate at which jumps of every possible size occur — a jump measure $\nu$ concentrated near zero with high total mass gives many tiny jumps (like a variance-gamma process); a $\nu$ with fat tails gives rare, large jumps (like Merton's jump-diffusion). Setting $\nu = 0$ recovers plain Brownian motion with drift; setting $\sigma = 0$ and using a constant-rate $\nu$ recovers a compound Poisson process.

The practical payoff is that any jump-diffusion option-pricing model — Merton, Kou, variance-gamma, CGMY — is really just a choice of Lévy measure $\nu$ plugged into this one template, so understanding the representation means you understand the entire model family at once rather than each model as a separate special case built from scratch.