École Nationale Supérieure de
Techniques Avancées

Abstract

This document studies the efficiency of the Wiener-Hopf (WH) simulation method for Lévy processes developped in Kuznetsov et al. [7]. This recent technique relies on the Wiener-Hopf factorisation for Lévy processes and on the simulation of the process at exponentially distributed times. The advantage of the method is that it allows to simulate simultaneously a process and its running supremum, which can have many applications, for instance in insurance (ruin probability and other related quantities for a Lévy risk process) and finance (pricing of exotic options in a Lévy model). First, this simulation technique is applied to the Brownian motion with drift, for which simulation errors can be estimated with closed formulae. So, this technique is compared to the classical one based on sampling increments of the process. Then, the Wiener-Hopf technique combined with a Monte-Carlo method (WHMC) is applied to the problem of optimal propotional reinsurance. This problem consists in minimising the ruin probability for an insurance risk Lévy process. In the last chapter, the limits of this Monte-Carlo minimisation and of the process simulation are examined.