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