On maxima and ladder processes for a dense class of Lévy processes

نویسنده

  • Martijn Pistorius
چکیده

Consider the problem to explicitly calculate the law of the first passage time T (a) of a general Lévy process Z above a positive level a. In this paper it is shown that the law of T (a) can be approximated arbitrarily closely by the laws of T (a), the corresponding first passages time for X, where (X)n is a sequence of Lévy processes whose positive jumps follow a phase-type distribution. Subsequently, explicit expressions are derived for the laws of T (a) and the upward ladder process of X. The derivation is based on an embedding of X into a class of Markov additive processes and on the solution of the fundamental (matrix) Wiener-Hopf factorisation for this class. This Wiener-Hopf factorisation can be computed explicitly by solving iteratively a certain fixed point equation. It is shown that, typically, this iteration converges geometrically fast.

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تاریخ انتشار 2008