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An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. (English) Zbl 1126.60039
Donati-Martin, Catherine (ed.) et al., Séminaire de Probabilités XL. Berlin: Springer (ISBN 978-3-540-71188-9/pbk). Lecture Notes in Mathematics 1899, 287-307 (2007).
Summary: Let $$X$$ be a spectrally negative Lévy process, reflect $$X$$ at its supremum $$\overline X$$ and call this process $$Y$$. Let $$\tau_a$$ denote the first time $$Y$$ crosses the level $$a$$. Using excursion theory we solve the problem of J. P. Lehoczky [Ann. Probab. 5, 601–607 (1977; Zbl 0367.60093)] or a spectrally negative Lévy process, that is, we express the joint law of $$(\tau_a,\overline X_{\tau_a},Y_{\tau_a-},\Delta X_{\tau_a})$$ in terms of so-called scale functions that also turn up in the solution of the two-sided exit problem, thereby extending results of F. Avram, A. E. Kyprianou and M. R. Pistorius [Ann. Appl. Probab. 14, No. 1, 215–238 (2004; Zbl 1042.60023)], who solved for the joint law of $$(\tau_a,Y_{\tau_a})$$. Next we obtain an explicit and non-randomised solution to the Skorokhod embedding problem of $$Y$$: we find a stopping time $$T$$ such that $$Y_T\sim\nu$$ for a measure $$\nu$$ on $$(0,\infty)$$ without atoms.
For the entire collection see [Zbl 1116.60002].

##### MSC:
 60G51 Processes with independent increments; Lévy processes
##### Keywords:
problem of Lehoczky; Skorokhod embedding
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