A Wiener.Hopf Monte Carlo simulation technique for Lévy processes

We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr.s so-called .Canadization. technique as well as Doney.s method of stochastic bounds for Lévy processes; see Carr [Rev. Fin. Studies 11 (1998) 597.626] and Doney [Ann. Probab. 32 (2004) 1545.1552]. We rely fundamentally on the Wiener.Hopf decomposition for Lévy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos Lévy en titillant la factorization de Wiener.Hopf (2002) Laboratoire de Mathématiques de L.INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801.1830]. We illustrate our Wiener.Hopf Monte Carlo method on a number of different processes, including a new family of Lévy processes called hypergeometric Lévy processes. Moreover, we illustrate the robustness of working with a Wiener.Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given Lévy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.