Skip to main content Accessibility help

A temporal approach to the Parisian risk model

  • Bin Li (a1), Gordon E. Willmot (a1) and Jeff T. Y. Wong (a1)


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).


Corresponding author

* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
** Email address:
*** Email address:
**** Email address:


Hide All
[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, 643656.
[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, 424452.
[3]Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 13641382.
[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, 97129.
[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, 572584.
[6]Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
[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, 13411377.
[8]Chesney, M. and Gauthier, L. (2006). American Parisian options. Finance Stoch. 10, 475506.
[9]Chesney, M., Jeanblanc-Picqué, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165184.
[10]Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk processes. J. Appl. Prob. 48, 9841002.
[11]Dai, M., Jiang, L. and Lin, J. (2013). Pricing corporate debt with finite maturity and chapter 11 proceedings. Quant. Finance 13, 18551861.
[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, 599615.
[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, 604620.
[14]Dassios, A. and Wu, S. (2008). Parisian ruin with exponential claims. Unpublished manuscript. Available at
[15]Dassios, A. and Wu, S. (2010). Perturbed Brownian motion and its application to Parisian option pricing. Finance Stoch. 14, 473494.
[16]Dassios, A. and Wu, S. (2011). Double-barrier Parisian options. J. Appl. Prob. 48, 120.
[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, 773804.
[18]Debnath, L. and Bhatta, D. (2015). Integral Transforms and Their Applications, 3rd edn. CRC, Boca Raton, FL.
[19]François, P. and Morellec, E. (2004). Capital structure and asset prices: some effects of bankruptcy procedures. J. Business 77, 387411.
[20]Galai, D., Raviv, A. and Wiener, Z. (2007). Liquidation triggers and the valuation of equity and debt. J. Banking Finance 31, 36043620.
[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. 97186.
[22]Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd edn. Springer, Heidelberg.
[23]Landriault, D., Renaud, J.-F. and Zhou, X. (2011). Occupation times of spectrally negative Lévy processes with applications. Stoch. Process. Appl. 121, 26292641.
[24]Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Prob. 16, 583607.
[25]Li, B. and Zhou, X. (2013). The joint Laplace transforms for diffusion occupation times. Adv. Appl. Prob. 45, 10491067.
[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.
[27]Lkabous, M. A., Czarna, I. and Renaud, J.-F. (2017). Parisian ruin for a refracted Lévy process. Insurance Math. Econom. 74, 153163.
[28]Loeffen, R., Czarna, I. and Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599609.
[29]Mejlbro, L. (2010). The Laplace Transformation I – General Theory: Complex Functions Theory a-4. Bookboon, London.
[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, 280290.


MSC classification

A temporal approach to the Parisian risk model

  • Bin Li (a1), Gordon E. Willmot (a1) and Jeff T. Y. Wong (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed