On error rates in normal approximations and simulation schemes for Lévy processes

Mikael Signahl

Abstract:: Let X = (X(t) : t /geq 0) be a Lévy process. In simulation, one often wants to know at what size it is possible to truncate the small jumps while retaining enough accuracy. A useful tool here is the Edgeworth expansion. We provide a third order expansion together with a uniform error bound, assuming third Lévy moment is 0. We next discuss approximating X in the finite variation case. Truncating the small jumps, we show that, adding their expected value, and further, include their variation, gives successively better results in general. Finally, some numerical illustrations involving a normal inverse Gaussian Lévy process are given.

Key words:: hidden Markov model, Markov-modulated Poisson process, traffic analysis, Poissonification, likelihood estimation, state space representation, Riccati equation, subspace identification