Li, Bin; Willmot, Gordon E.; Wong, Jeff T. Y. A temporal approach to the Parisian risk model. (English) Zbl 1396.60045 J. Appl. Probab. 55, No. 1, 302-317 (2018). Summary: In this paper, we propose a new approach to study the Parisian ruin problem for spectrally negative Lévy processes. Since our approach is based on a hybrid observation scheme switching between discrete and continuous observations, we call it a temporal approach as opposed to the spatial approximation approach in the literature. Our approach leads to a unified proof for the underlying processes with bounded or unbounded variation paths, and our result generalizes [R. Loeffen et al., Bernoulli 19, No. 2, 599–609 (2013; Zbl 1267.60054)]. Cited in 2 ReviewsCited in 8 Documents MSC: 60G40 Stopping times; optimal stopping problems; gambling theory 60G51 Processes with independent increments; Lévy processes Keywords:Parisian ruin; Poisson observation; insurance risk model; spectrally negative Lévy process Citations:Zbl 1267.60054 PDFBibTeX XMLCite \textit{B. Li} et al., J. Appl. Probab. 55, No. 1, 302--317 (2018; Zbl 1396.60045) Full Text: DOI References: [1] Albrecher, H.; Ivanovs, J., Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations, Stoch. Process. Appl., 127, 643-656, (2017) · Zbl 1354.60048 [2] Albrecher, H.; Cheung, E. C. K.; Thonhauser, S., Randomized observation periods for the compound Poisson risk model: the discounted penalty function, Scand. Actuarial J., 2013, 424-452, (2013) · Zbl 1401.91089 [3] Albrecher, H.; Ivanovs, J.; Zhou, X., Exit identities for Lévy processes observed at Poisson arrival times, Bernoulli, 22, 1364-1382, (2016) · Zbl 1338.60125 [4] Albrecher, H.; Kortschak, D.; Zhou, X., Pricing of Parisian options for a jump-diffusion model with two-sided jumps, Appl. Math. Finance, 19, 97-129, (2012) · Zbl 1372.91100 [5] Baurdoux, E. J.; Pardo, J. C.; Pérez, J. L.; Renaud, J.-F., Gerber-Shiu distribution at Parisian ruin for Lévy insurance risk processes, J. Appl. Prob., 53, 572-584, (2016) · Zbl 1344.60046 [6] Bertoin, J., Lévy Processes, (1996), Cambridge University Press [7] Broadie, M.; Chernov, M.; Sundaresan, S., Optimal debt and equity values in the presence of Chapter 7 and Chapter 11, J. Finance, 62, 1341-1377, (2007) [8] Chesney, M.; Gauthier, L., American Parisian options, Finance Stoch., 10, 475-506, (2006) · Zbl 1126.91025 [9] Chesney, M.; Jeanblanc-Picqué, M.; Yor, M., Brownian excursions and Parisian barrier options, Adv. Appl. Prob., 29, 165-184, (1997) · Zbl 0882.60042 [10] Czarna, I.; Palmowski, Z., Ruin probability with Parisian delay for a spectrally negative Lévy risk processes, J. Appl. Prob., 48, 984-1002, (2011) · Zbl 1232.60036 [11] Dai, M.; Jiang, L.; Lin, J., Pricing corporate debt with finite maturity and chapter 11 proceedings, Quant. Finance, 13, 1855-1861, (2013) · Zbl 1282.91368 [12] Dassios, A.; Lim, J. W., Parisian option pricing: a recursive solution for the density of the Parisian stopping time, SIAM J. Financial Math., 4, 599-615, (2013) · Zbl 1295.91098 [13] Dassios, A.; Lim, J. W., An analytical solution for the two-sided Parisian stopping time, its asymptotics, and the pricing of Parisian options, Math. Finance, 27, 604-620, (2017) [14] Dassios, A.; Wu, S., Parisian ruin with exponential claims, (2008) [15] Dassios, A.; Wu, S., Perturbed Brownian motion and its application to Parisian option pricing, Finance Stoch., 14, 473-494, (2010) · Zbl 1226.91073 [16] Dassios, A.; Wu, S., Double-barrier Parisian options, J. Appl. Prob., 48, 1-20, (2011) · Zbl 1208.91143 [17] Dassios, A.; Zhang, Y. Y., The joint distribution of Parisian and hitting times of Brownian motion with application to Parisian option pricing, Finance Stoch., 20, 773-804, (2016) · Zbl 1369.91176 [18] Debnath, L.; Bhatta, D., Integral Transforms and Their Applications, (2015), CRC: CRC, Boca Raton, FL · Zbl 1310.44001 [19] François, P.; Morellec, E., Capital structure and asset prices: some effects of bankruptcy procedures, J. Business, 77, 387-411, (2004) [20] Galai, D.; Raviv, A.; Wiener, Z., Liquidation triggers and the valuation of equity and debt, J. Banking Finance, 31, 3604-3620, (2007) [21] Kuznetsov, A.; Kyprianou, A. E.; Rivero, V., Lévy Matters II, The theory of scale functions for spectrally negative Lévy processes, 97-186, (2012), Springer: Springer, Heidelberg · Zbl 1261.60047 [22] Kyprianou, A. E., Fluctuations of Lévy Processes with Applications: Introductory Lectures, (2014), Springer: Springer, Heidelberg · Zbl 1384.60003 [23] Landriault, D.; Renaud, J.-F.; Zhou, X., Occupation times of spectrally negative Lévy processes with applications, Stoch. Process. Appl., 121, 2629-2641, (2011) · Zbl 1227.60061 [24] Landriault, D.; Renaud, J.-F.; Zhou, X., An insurance risk model with Parisian implementation delays, Methodol. Comput. Appl. Prob., 16, 583-607, (2014) · Zbl 1319.60098 [25] Li, B.; Zhou, X., The joint Laplace transforms for diffusion occupation times, Adv. Appl. Prob., 45, 1049-1067, (2013) · Zbl 1370.60136 [26] Li, B.; Tang, Q.; Wang, L.; Zhou, X., Liquidation risk in the presence of Chapters 7 and 11 of the US bankruptcy code, J. Financial Eng., 1, (2014) [27] Lkabous, M. A.; Czarna, I.; Renaud, J.-F., Parisian ruin for a refracted Lévy process, Insurance Math. Econom., 74, 153-163, (2017) · Zbl 1394.60046 [28] Loeffen, R.; Czarna, I.; Palmowski, Z., Parisian ruin probability for spectrally negative Lévy processes, Bernoulli, 19, 599-609, (2013) · Zbl 1267.60054 [29] Mejlbro, L., The Laplace Transformation I - General Theory: Complex Functions Theory a-4, (2010), Bookboon: Bookboon, London [30] Wong, J. T. Y.; Cheung, E. C. K., On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps, Insurance Math. Econom., 65, 280-290, (2015) · Zbl 1348.91189 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.