Weak reflection principle for spectrally negative L\'evy processes
AbstractIn this paper, we develop a new mathematical technique which can be used to express the joint distribution of a Markov process and its running maximum (or minimum) through the distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the standard reflection principle to work. We call this method a weak reflection principle and show that it provides solutions to many problems for which the classical reflection principle is typically used. In addition, unlike the standard reflection principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak reflection principle for a large class of time-homogeneous diffusions on a real line and, then, proceed to develop this method for all L\'evy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak reflection principle in Financial Mathematics, Computational Methods, and Inverse Problems.
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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1308.2250.
Date of creation: Aug 2013
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Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-08-16 (All new papers)
- NEP-ECM-2013-08-16 (Econometrics)
- NEP-SPO-2013-08-16 (Sports & Economics)
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