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The Lévy-Khintchine Formula: Characterizing Jump-Diffusion 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) is uniquely determined by the triplet (A,ν,γ) (A, \nu, \gamma) via the Lévy-Khintchine representation:
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, Au \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)
where A A is a symmetric non-negative definite matrix, ν \nu is a Lévy measure satisfying ∫Rd∖{0}min⁡(1,∣x∣2)ν(dx)<∞ \int_{\mathbb{R}^d \setminus \{0\}} \min(1, |x|^2) \nu(dx) < \infty , and γ∈Rd \gamma \in \mathbb{R}^d is the drift vector.

Analytical Intuition.

Imagine a particle traversing a landscape, subject to three distinct cosmic forces. First, the γ \gamma term represents a steady, deterministic drift—the relentless current pushing the particle forward. Second, the A A matrix embodies the Brownian component, a jittery, continuous diffusion analogous to thermal agitation in a fluid. Finally, the integral against the Lévy measure ν \nu accounts for the 'catastrophic' discontinuities: the jumps. Think of these as sudden quantum leaps or seismic shocks where the particle vanishes from one position and instantly reappears elsewhere. The beauty of the Lévy-Khintchine formula lies in its role as a spectral DNA sequence; by observing only the characteristic function of the process, we can perfectly decompose any complex stochastic movement into these three fundamental components. It provides a universal language to unify Brownian motion, Poisson processes, and stable distributions into a single, rigorous framework, revealing the hidden anatomy of all processes with stationary and independent increments.
CAUTION

Institutional Warning.

Students often struggle with the truncation function 1∣x∣<1 \mathbb{1}_{|x|<1} . It is required for convergence when jumps have infinite variation; it effectively 'subtracts' the small, frequent oscillations to prevent the integral from diverging to infinity, ensuring the formula remains well-defined for all u u .

Academic Inquiries.

01

Why is the triplet (A,ν,γ) (A, \nu, \gamma) unique?

Because the characteristic exponent ψ(u) \psi(u) is the logarithm of the Fourier transform of the distribution; the uniqueness follows from the injectivity of the Fourier-Stieltjes transform.

02

What happens if ν=0 \nu = 0 ?

The process reduces to a Brownian motion with drift, as the jump component is entirely silenced, leaving only the Gaussian diffusion.

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). The Lévy-Khintchine Formula: Characterizing Jump-Diffusion Processes: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-l-vy-khintchine-formula--characterizing-jump-diffusion-processes

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