Approximations of small jumps of Lévy processes with a view towards simulation

Abstract

Let X = (X(t):t ≥ 0) be a Lévy process and X_𝜀 the compensated sum of jumps not exceeding 𝜀 in absolute value, 𝜎²(𝜀) = var(X_𝜀(1)). In simulation, X - X_𝜀 is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X_𝜀/𝜎(𝜀) can be approximated by another Brownian term. A necessary and sufficient condition in terms of 𝜎(𝜀) is given, and it is shown that when the condition fails, the behaviour of X_𝜀/𝜎(𝜀) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.