MPS-RR 2000-26
July 2000

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

Søren Asmussen

Jan Rosinski

Abstract

Let $X=(X(t):, t\ge 0)$ be a L'evy process and $X_\epsilon$ the compensated sum of jumps not exceeding $\ep$ in absolute value, $\sigma^2(\ep)=$ $\Var(X_\ep(1))$. In simulation, $X-X_\ep$ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when $X_\ep/\sigma(\ep)$ can be approximated by another Brownian term. A necessary and sufficient condition in terms of $\sigma(\ep)$ is given, and it is shown that when the condition fails, the behaviour of $X_\ep/\sigma(\ep)$ 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.

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