An explicit Skorokhod embedding for spectrally negative Levy processes

Details An explicit Skorokhod embedding for spectrally negative Levy processes An explicit Skorokhod embedding for spectrally negative Levy processes

2007-03-20

arxiv.org/abs/math/0703597v3

Mathematics

Probability

This is the final version of the paper that has been accepted for publication in J. Theor. Probab. In this new version several

Scientific paper

10.1007/s10959-008-0157-7

We present an explicit solution to the Skorokhod embedding problem for spectrally negative L\'evy processes. Given a process $X$ and a target measure $\mu$ satisfying an explicit admissibility condition we define functions $\f_\pm$ such that the stopping time $T = \inf\{t>0: X_t \in \{-\f_-(L_t), \f_+(L_t)\}\}$ induces $X_T\sim \mu$. We also treat versions of $T$ which take into account the sign of the excursion straddling time $t$. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (1992) and we compute explicitly their general quantities in our setup. Our method relies on some new explicit calculations relating scale functions and the It\^o excursion measure of $X$. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.

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