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Lewis and Mordecki have computed the Wiener-Hopf factorization of a L\'evyprocess whose restriction on $]0,+\infty[$ of their L\'evy measure has arational Laplace transform. That allows to compute the distribution of$(X_t,\inf_{0\leq s\leq t}X_s)$. For the same class of L\'evy processes, wecompute the distribution of $ (X_t,\inf_{0\leq s\leq t}X_s,\sup_{0\leq s\leq t}X_s)$ and also the behavior of this triple at certain stopping time, like thefirst exit time of an interval containing the origin. Some applications to thepricing of double barrier options with or without rebate are evocated.

Explicit solutions for the exit problem for a class of L\'evy processes. Applications to the pricing of double barrier options