LévyKhintchine

LévyKhintchine refers to the Lévy–Khintchine representation, a foundational result in probability theory that describes the characteristic functions of infinitely divisible distributions and underpins Lévy processes. The representation expresses a distribution as a combination of drift, diffusion, and jumps, encoded in a Lévy triplet.

In d dimensions, for a characteristic exponent ψ(t) with t in R^d, the Lévy–Khintchine formula states

ψ(t) = i ⟨t, γ⟩ − 1/2 ⟨t, A t⟩ + ∫_{R^d} (e^{i⟨t, x⟩} − 1 − i⟨t, x⟩ 1_{|x|<1}) ν(dx),

where γ ∈ R^d is a drift vector, A is a nonnegative definite d×d matrix representing Gaussian diffusion, and ν is the Lévy measure describing jumps. The measure ν satisfies ∫_{R^d} (|x|^2 ∧ 1) ν(dx) < ∞. The characteristic function φ(t) of the distribution is φ(t) = exp(ψ(t)).

In one dimension, the formula specializes to

ψ(t) = i t γ − 1/2 σ^2 t^2 + ∫ (e^{i t x} − 1 − i t x 1_{|x|<1}) ν(dx),

with ν meeting the same integrability condition.

For a Lévy process X_t, the characteristic function satisfies E[e^{i⟨t, X_t⟩}] = exp(t ψ(t)), highlighting the time-homogeneous, stationary independent increments structure. The Lévy triplet (γ, A, ν) fully characterizes the process: γ captures drift, A the Brownian component, and ν the jump part through the jump measure.

Applications include modeling with jumps in finance, physics, and risk theory, where the framework separates continuous and discontinuous movements.