Lévy Processes

Core Idea

A Lévy process is a stochastic process with stationary and independent increments and càdlàg (right-continuous with left limits) sample paths. The Lévy-Khintchine formula characterizes its distribution through three components: a deterministic drift b, a Gaussian (Brownian) part with variance σ², and a jump part described by a Lévy measure ν. Every Lévy process decomposes as X(t) = bt + σW(t) + J(t), where J captures all jumps. This unifies Brownian motion (ν = 0), Poisson processes (σ = 0, ν atomic), and their combinations.

Explainer

Lévy processes are the natural generalization of Brownian motion and Poisson processes to a unified framework. A Lévy process X(t) has three defining properties: X(0) = 0, stationary increments (X(t+s) - X(t) has the same distribution as X(s)), independent increments, and càdlàg paths (right-continuous with left limits, allowing jumps). Brownian motion satisfies these with continuous paths; the Poisson process satisfies them with piecewise-constant paths. Lévy processes include both extremes and everything in between — processes that simultaneously diffuse continuously and jump randomly.

The Lévy-Khintchine formula provides a complete classification. Every Lévy process is determined by a characteristic triplet (b, σ², ν), where b ∈ ℝ is a drift, σ² ≥ 0 is the Brownian variance, and ν is a Lévy measure on ℝ\{0} satisfying ∫min(1, x²)ν(dx) < ∞. The characteristic exponent ψ(u) = log E[e^iuX(1)] = ibu - σ²u²/2 + ∫(e^iux - 1 - iux·1_{|x|<1})ν(dx) uniquely determines the distribution. This formula is the continuous-time analogue of the fact that infinitely divisible distributions are classified by Gaussian and Poisson components. The condition ∫min(1,x²)ν(dx) < ∞ allows ν to have infinite total mass (infinitely many small jumps per unit time) but requires the small jumps to be square-integrable.

The Lévy-Itô decomposition provides the pathwise structure. Every Lévy process decomposes as the independent sum of: (1) a deterministic drift bt, (2) a Brownian motion σW(t), (3) a compound Poisson process of large jumps (|x| ≥ 1, occurring at finite rate), and (4) a compensated sum of small jumps (|x| < 1, centered to have zero mean). This decomposition is canonical — there is no other type of behavior a process with stationary independent increments can exhibit. It is the continuous-time analogue of the decomposition of an infinitely divisible random variable into Gaussian and Poisson components.

Important examples beyond Brownian motion and Poisson processes include: the compound Poisson process (finite Lévy measure, finitely many jumps per unit time), the Cauchy process (stable process with index 1, ν(dx) = c/|x|² dx, infinite jump activity), the variance-gamma process (a Brownian motion with drift time-changed by a gamma process, used in finance for heavy-tailed returns), and stable processes (self-similar Lévy processes, the continuous-time analogues of stable distributions). In mathematical finance, Lévy processes address the key deficiency of geometric Brownian motion: they can produce heavy tails, skewness, and discontinuous price movements (jumps), all observed in real market data. The stochastic calculus for Lévy processes extends Itô calculus by adding a jump integral ∫∫f(x)(N(dt,dx) - ν(dx)dt) against the compensated Poisson random measure.