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Lévy-Khintchine Representation for Lévy Processes

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The Formal Theorem

Let X={Xt:t≥0} X = \{X_t : t \ge 0\} be a Lévy process on Rd \mathbb{R}^d . The characteristic function of Xt X_t is given by E[ei⟨u,Xt⟩]=etΨ(u) E[e^{i \langle u, X_t \rangle}] = e^{t \Psi(u)} , where the characteristic exponent Ψ(u) \Psi(u) satisfies:
Ψ(u)=i⟨γ,u⟩−12⟨u,Au⟩+∫Rd∖{0}(ei⟨u,x⟩−1−i⟨u,x⟩1{∣x∣<1})ν(dx) \Psi(u) = i \langle \gamma, u \rangle - \frac{1}{2} \langle u, A u \rangle + \int_{\mathbb{R}^d \setminus \{0\}} \left( e^{i \langle u, x \rangle} - 1 - i \langle u, x \rangle \mathbb{1}_{\{|x|<1\}} \right) \nu(dx)
Here, γ∈Rd \gamma \in \mathbb{R}^d is the drift vector, A A is a symmetric non-negative definite d×d d \times d matrix (diffusion part), and ν \nu is a Lévy measure on Rd∖{0} \mathbb{R}^d \setminus \{0\} satisfying ∫Rd∖{0}min⁡(1,∣x∣2)ν(dx)<∞ \int_{\mathbb{R}^d \setminus \{0\}} \min(1, |x|^2) \nu(dx) < \infty .

Analytical Intuition.

Imagine the path of a particle as a complex choreography. The Ψ(u) \Psi(u) function acts as the 'DNA' of the process, uniquely encoding its entire probabilistic behavior at any time t t . The representation decomposes this motion into three distinct physical forces. First, the drift γ \gamma represents a deterministic, linear translation. Second, the matrix A A characterizes the 'smooth' Gaussian-like fluctuations, akin to Brownian motion. Finally, the integral term acts as the 'quantum' component: the Lévy measure ν \nu catalogs the frequency and size of jumps. Small jumps (near the origin) are bundled together to ensure the process remains well-behaved, while large jumps are counted individually. By summing these components, the Lévy-Khintchine formula proves that any process with stationary, independent increments is fundamentally a hybrid of a linear trend, a diffusion, and a jump-counting machine. It is the definitive 'map' of all possible stochastic continuity and disruption.
CAUTION

Institutional Warning.

Students often conflate the Lévy measure ν(dx) \nu(dx) with a probability distribution. It is not; it is a measure that can be infinite near the origin. The compensation term −i⟨u,x⟩1{∣x∣<1} - i \langle u, x \rangle \mathbb{1}_{\{|x|<1\}} is essential to ensure the integral converges, essentially 'centering' the small jumps.

Academic Inquiries.

01

Why do we use the indicator function in the integral?

The indicator function 1{∣x∣<1} \mathbb{1}_{\{|x|<1\}} acts as a compensator for small jumps. Without it, the integral might diverge near zero, preventing the existence of the characteristic exponent.

02

Can a Lévy process have only jumps and no diffusion?

Yes. A Pure Jump Lévy process occurs when A=0 A = 0 . Examples include the Poisson process or more complex processes like the Variance Gamma process.

Standardized References.

  • Definitive Institutional SourceApplebaum, D., Lévy Processes and Stochastic Calculus.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Lévy-Khintchine Representation for Lévy Processes: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/l-vy-khintchine-representation-for-l-vy-processes

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