Explicit solutions of the exit problem for a class of Lévy processes; applications to the pricing of double-barrier options
Résumé
Lewis and Mordecki have computed the Wiener-Hopf factorization of a Levy process whose restriction of the Levy measure on vertical bar 0, +infinity vertical bar has a rational Laplace transform. This allowed them to compute the distribution of (X-t, inf(0 <= s <= t) X-s). For the same class of Levy processes, we compute the distribution of
(X-t, inf(0 <= s <= t) X-s, sup(0 <= s <= t) X-s.)
and also the behavior of this triple at certain stopping times, such as the time of first exit of an interval containing the origin. Some applications to the pricing of double-barrier options with or without rebate are described.