The Lévy-Khintchine formula states that the characteristic exponent ψ(u) = log E[e^{iuX(1)}] of a Lévy process has the form ψ(u) = ibu - σ²u²/2 + ∫(e^{iux} - 1 - iux·1_{|x|<1}) ν(dx). The three terms correspond to:
CReal part, imaginary part, and modulus of the characteristic function
DSmall jumps, medium jumps, and large jumps
The Lévy-Khintchine formula is the complete classification of infinitely divisible distributions: ibu is a deterministic drift, -σ²u²/2 is the Gaussian (Brownian) component, and the integral captures all jump activity through the Lévy measure ν. The measure ν(dx) describes the rate of jumps of size x: ν(A) is the expected number of jumps per unit time whose size falls in the set A. The truncation function 1_{|x|<1} compensates for small jumps (which may be so frequent that their sum needs centering). Every Lévy process is uniquely determined by the triplet (b, σ², ν).
Question 2 Multiple Choice
Brownian motion is a Lévy process with (b, σ², ν) = (0, 1, 0) and a Poisson process with rate λ has (b, σ², ν) = (0, 0, λδ₁). A compound Poisson process with jump rate λ and jump distribution F has Lévy measure:
Aν(dx) = λF(dx), concentrating ν on the jump-size distribution weighted by the jump rate
Bν(dx) = λδ₀(dx), concentrating all mass at zero
Cν(dx) = F(dx)/λ, inversely proportional to the rate
Dν(dx) = λ²F(dx), squared because the process is compound
A compound Poisson process has jumps arriving at rate λ, each with size drawn from F. The Lévy measure ν(dx) = λF(dx) encodes both the rate and size distribution: the total mass ν(ℝ\{0}) = λ is the jump rate, and the normalized measure F(dx) = ν(dx)/λ gives the jump size distribution. Key property: compound Poisson processes have ν(ℝ\{0}) < ∞ (finite jump rate). Lévy processes with ∫ν(dx) = ∞ (like the Cauchy process or variance-gamma process) have infinitely many jumps in every interval — a qualitatively different behavior.
Question 3 Short Answer
Explain the Lévy-Itô decomposition and why it is the 'structure theorem' for Lévy processes.
Think about your answer, then reveal below.
Model answer: The Lévy-Itô decomposition writes every Lévy process as the independent sum of three components: X(t) = bt + σW(t) + X^{large}(t) + X^{small}(t). The first is deterministic drift, the second is a Brownian motion (continuous Gaussian part), and the third and fourth capture jumps. Large jumps (|x| ≥ 1) form a compound Poisson process X^{large} (finitely many per unit time). Small jumps (|x| < 1) are handled by a compensated sum X^{small} = lim_{ε→0} (sum of jumps in (ε,1) minus their mean) — this converges because the compensation removes the potentially divergent mean. The decomposition shows that drift, diffusion, and jumps are the three and only three types of behavior a Lévy process can exhibit.
The decomposition is 'structural' because it is exhaustive and canonical: there is no fourth type of behavior. Any process with stationary independent increments must be a combination of these three. This is the content of the Lévy-Khintchine theorem — the triplet (b, σ², ν) parametrizes all possibilities.