Abstract
The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Lévy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also provided. The existing formulas for Brownian motion with drift are recovered using the particular scale functions.
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Avram, F., Kyprianou, A.E., Pistorius, M.R.: Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14, 215–238 (2004)
Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (2007)
Çinlar, E.: Probability and Stochastics. Springer, New York (2011)
Hubalek, F., Kyprianou, A.E.: Old and new examples of scale functions for spectrally negative Lévy processes. Progress in Probability 63, 119–145 (2010)
Kuznetsov, A., Kyprianou, A.E., Rivero, V.: The theory of scale functions for spectrally negative Lévy processes, Lévy matters II. Lect. Notes Math. 2061, 97–186 (2013)
Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, 2nd Edn. Springer, New York (2014)
Mijatovic, A., Pistorius, M.R.: On the drawdown of completely asymmetric Lévy processes. Stoch. Proc. Appl. 122, 3812–3836 (2012)
Pistorius, M.R.: On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theor. Probab. 17, 183–220 (2004)
Salminen, P., Vallois, P.: On maximum increase and decrease of Brownian motion. Ann. I. H. Poincaré 43, 655–676 (2007)
Vardar-Acar, C., Zirbel, C.L., Székely, G.J.: On the correlation of the supremum and the infimum and of maximum gain and maximum loss of Brownian motion with drift. J. Comput. Appl. Math. 248, 61–75 (2013)
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This work is supported by the Scientific and Technological Research Council of Turkey, TUBITAK Project No.110T674
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Vardar-Acar, C., Çağlar, M. Maximum loss and maximum gain of spectrally negative Lévy processes. Extremes 20, 301–308 (2017). https://doi.org/10.1007/s10687-016-0279-8
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DOI: https://doi.org/10.1007/s10687-016-0279-8