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Levy Process

The general class of continuous-time processes with stationary, independent increments and right-continuous paths, encompassing Brownian motion and the Poisson process as the two purest special cases, along with a wide family of jump-driven processes in between.

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In Plain English

A Levy process is what you get from insisting on just two properties, stationary increments (the statistical behavior of a change over a given time span doesn't depend on when that span starts) and independent increments (non-overlapping changes are independent of each other), without insisting on continuous paths the way Brownian motion does. This turns out to be a remarkably rich family: Brownian motion (pure continuous diffusion, no jumps) and the Poisson process (pure jumps, no continuous drift) sit at opposite ends of the same spectrum, and the Levy-Khintchine formula shows that every Levy process is some combination of a continuous Brownian part plus a jump part, making 'Levy process' the umbrella covering both alongside everything in between.

Definition

A Levy process is a stochastic process ${X_{t}}_{t \geq 0}$ with $X_{0} = 0$ satisfying: (1) independent increments, (2) stationary increments (the distribution of $X_{t + s} - X_{t}$ depends only on $s$ , not on $t$ ), and (3) stochastic continuity (for any $ϵ > 0$ , $P (∣ X_{t + h} - X_{t} ∣ > ϵ) \to 0$ as $h \to 0$ , ruling out fixed, scheduled jump times but permitting random jumps). Every Levy process has a càdlàg (right-continuous with left limits) modification, and the Levy-Khintchine formula characterizes its characteristic function entirely via a triplet $(b, σ^{2}, ν)$ : a drift term $b$ , a Gaussian (Brownian) diffusion component with variance $σ^{2}$ , and a jump measure $ν$ governing the frequency and size distribution of jumps. Brownian motion (with drift) is the pure-diffusion Levy process ( $ν = 0$ ); the Poisson process and compound Poisson process are pure-jump Levy processes ( $σ^{2} = 0$ ); more general Levy processes combine both continuous diffusion and jumps, used to model, for example, sudden price jumps in financial asset models that geometric Brownian motion cannot capture.

Properties

Brownian motion and the Poisson process are often taught as unrelated processes, one continuous, one a jump-counting process, the Levy process framework reveals they are actually the two extreme special cases of a single unified family, differing only in which piece of the Levy-Khintchine triplet is zero.
This entry is a distinct concept from the Levy distribution (a specific, particular heavy-tailed stable probability distribution): a Levy process is the general class of processes with stationary independent increments, its increments need not themselves follow a Levy distribution, that connection only holds for the specific case of a Levy stable process built from Levy-distributed increments.
Financial models needing sudden, discontinuous price jumps (crashes, earnings surprises) that geometric Brownian motion cannot produce often turn to jump-diffusion Levy processes, which combine a continuous Brownian component with a compound Poisson jump component, still within the same Levy-Khintchine framework.

At a Glance

Defining propertiesstationary and independent increments, plus stochastic continuity

Levy-Khintchine triplet

(b, σ^{2}, ν)

: drift, Gaussian variance, and jump measure

Pure-diffusion special caseBrownian motion (with drift), no jump component

Pure-jump special casethe Poisson and compound Poisson processes, no diffusion component

Last updated September 8, 2026← Back to the Dictionary