Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-23T22:32:08.703Z Has data issue: false hasContentIssue false
  • English
  • Français

A temporal approach to the Parisian risk model

Part of: Stochastic processes

Published online by Cambridge University Press:  28 March 2018

Bin Li ,
Gordon E. Willmot  and
Jeff T. Y. Wong
Bin Li*
Affiliation:
University of Waterloo
Gordon E. Willmot*
Affiliation:
University of Waterloo
Jeff T. Y. Wong*
Affiliation:
University of Waterloo
*
* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

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 Loeffen et al. (2013).

Keywords

Parisian ruinPoisson observationinsurance risk modelspectrally negative Lévy process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G51: Processes with independent increments; L'evy processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 302 - 317
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Albrecher, H. and Ivanovs, J. (2017). Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations. Stoch. Process. Appl. 127, 643–656. CrossRefGoogle Scholar
[2]Albrecher, H., Cheung, E. C. K. and Thonhauser, S. (2013). Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scand. Actuarial J. 2013, 424–452. CrossRefGoogle Scholar
[3]Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 1364–1382. Google Scholar
[4]Albrecher, H., Kortschak, D. and Zhou, X. (2012). Pricing of Parisian options for a jump-diffusion model with two-sided jumps. Appl. Math. Finance 19, 97–129. CrossRefGoogle Scholar
[5]Baurdoux, E. J., Pardo, J. C., Pérez, J. L. and Renaud, J.-F. (2016). Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes. J. Appl. Prob. 53, 572–584. Google Scholar
[6]Bertoin, J. (1996). Lévy Processes. Cambridge University Press. Google Scholar
[7]Broadie, M., Chernov, M. and Sundaresan, S. (2007). Optimal debt and equity values in the presence of Chapter 7 and Chapter 11. J. Finance 62, 1341–1377. Google Scholar
[8]Chesney, M. and Gauthier, L. (2006). American Parisian options. Finance Stoch. 10, 475–506. CrossRefGoogle Scholar
[9]Chesney, M., Jeanblanc-Picqué, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165–184. CrossRefGoogle Scholar
[10]Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk processes. J. Appl. Prob. 48, 984–1002. Google Scholar
[11]Dai, M., Jiang, L. and Lin, J. (2013). Pricing corporate debt with finite maturity and chapter 11 proceedings. Quant. Finance 13, 1855–1861. Google Scholar
[12]Dassios, A. and Lim, J. W. (2013). Parisian option pricing: a recursive solution for the density of the Parisian stopping time. SIAM J. Financial Math. 4, 599–615. Google Scholar
[13]Dassios, A. and Lim, J. W. (2017). An analytical solution for the two-sided Parisian stopping time, its asymptotics, and the pricing of Parisian options. Math. Finance 27, 604–620. Google Scholar
[14]Dassios, A. and Wu, S. (2008). Parisian ruin with exponential claims. Unpublished manuscript. Available at http://stats.lse.ac.uk/angelos/. Google Scholar
[15]Dassios, A. and Wu, S. (2010). Perturbed Brownian motion and its application to Parisian option pricing. Finance Stoch. 14, 473–494. Google Scholar
[16]Dassios, A. and Wu, S. (2011). Double-barrier Parisian options. J. Appl. Prob. 48, 1–20. CrossRefGoogle Scholar
[17]Dassios, A. and Zhang, Y. Y. (2016). The joint distribution of Parisian and hitting times of Brownian motion with application to Parisian option pricing. Finance Stoch. 20, 773–804. Google Scholar
[18]Debnath, L. and Bhatta, D. (2015). Integral Transforms and Their Applications, 3rd edn. CRC, Boca Raton, FL. Google Scholar
[19]François, P. and Morellec, E. (2004). Capital structure and asset prices: some effects of bankruptcy procedures. J. Business 77, 387–411. Google Scholar
[20]Galai, D., Raviv, A. and Wiener, Z. (2007). Liquidation triggers and the valuation of equity and debt. J. Banking Finance 31, 3604–3620. Google Scholar
[21]Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II, Springer, Heidelberg, pp. 97–186. Google Scholar
[22]Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd edn. Springer, Heidelberg. CrossRefGoogle Scholar
[23]Landriault, D., Renaud, J.-F. and Zhou, X. (2011). Occupation times of spectrally negative Lévy processes with applications. Stoch. Process. Appl. 121, 2629–2641. Google Scholar
[24]Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Prob. 16, 583–607. CrossRefGoogle Scholar
[25]Li, B. and Zhou, X. (2013). The joint Laplace transforms for diffusion occupation times. Adv. Appl. Prob. 45, 1049–1067. Google Scholar
[26]Li, B., Tang, Q., Wang, L. and Zhou, X. (2014). Liquidation risk in the presence of Chapters 7 and 11 of the US bankruptcy code. J. Financial Eng. 1, 1450023. Google Scholar
[27]Lkabous, M. A., Czarna, I. and Renaud, J.-F. (2017). Parisian ruin for a refracted Lévy process. Insurance Math. Econom. 74, 153–163. Google Scholar
[28]Loeffen, R., Czarna, I. and Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599–609. Google Scholar
[29]Mejlbro, L. (2010). The Laplace Transformation I – General Theory: Complex Functions Theory a-4. Bookboon, London. Google Scholar
[30]Wong, J. T. Y. and Cheung, E. C. K. (2015). On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps. Insurance Math. Econom. 65, 280–290. CrossRefGoogle Scholar