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 L\'evy 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 L\'evy processes. We rely fundamentally on the Wiener-Hopf decomposition for L\'evy processes as well as taking advantage of recent developments in factorisation techniques of the latter theory due to Vigon and Kuznetsov. We illustrate our Wiener-Hopf Monte-Carlo method on a number of different processes, including a new family of L\'evy processes called hypergeometric L\'evy 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 L\'evy 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.