A Wiener--Hopf Monte Carlo simulation technique for L\'evy processes
Abstract
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; 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 L\'evy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos L\'evy en titillant la factorization de Wiener-Hopf (2002) Laboratoire de Math\'ematiques 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 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.
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