Let X be a spectrally negative Lévy process, reflect X at its supremum X and call this process Y. Let ?a denote the first time Y crosses the level a. Using excursion theory we solve the problem of Lehoczky for a spectrally negative Lévy process, that is, we express the joint law of (Ta, XTa, YTa-ΔXTa ) in terms of socalled scale functions that also turn up in the solution of the two-sided exit problem, thereby extending results of Avram et al. [2], who solved for the joint law of (Ta, YTa ). Next we obtain an explicit and non-randomised solution to the Skorokhod embedding problem of Y : we find a stopping time T such that YT ∼ ? for a measure ? on (0,∞) without atoms. Key words: Lévy process, Ito excursion theory, First passage, Skorokhod embedding, Problem of Lehoczky
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© 2007 Springer-VerlagBerlinHeidelberg
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Pistorius, M.R. (2007). An Excursion-Theoretical Approach to Some Boundary Crossing Problems and the Skorokhod Embedding for Reflected Lévy Processes. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XL. Lecture Notes in Mathematics, vol 1899. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71189-6_15
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DOI: https://doi.org/10.1007/978-3-540-71189-6_15
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