Skip to main content Accessibility help
×
×
Home

The distribution of refracted Lévy processes with jumps having rational Laplace transforms

  • Jiang Zhou (a1) and Lan Wu (a1)

Abstract

We consider a refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulae for the Laplace transform of its distribution. Our formulae are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. In addition, the results are used to price variable annuities with state-dependent fees.

Copyright

Corresponding author

* Postal address: School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
** Email address: 1101110056@pku.edu.cn
*** Email address: lwu@pku.edu.cn

References

Hide All
[1] Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Prob. 15, 20622080.
[2] Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79111.
[3] Bauer, D., Kling, A. and Russ, J. (2008). A universal pricing framework for guaranteed minimum benefits in variable annuities. ASTIN Bull. 38, 621651.
[4] Bernard, C., Hardy, M. and MacKay, A. (2014). State-dependent fees for variable annuity guarantees. ASTIN Bull. 44, 559585.
[5] Cai, N. (2009). On first passage times of a hyper-exponential jump diffusion process. Operat. Res. Lett. 37, 127134.
[6] Cai, N., Chen, N. and Wan, X. (2009). Pricing double-barrier options under a flexible jump diffusion model. Operat. Res. Lett. 37, 163167.
[7] Delong, Ł. (2014). Pricing and hedging of variable annuities with state-dependent fees. Insurance Math. Econom. 58, 2433.
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
[9] Gerber, H. U. and Shiu, E. S. W. (1994). Option pricing by Esscher transforms. Trans. Soc. Actuaries 46, 99140.
[10] Ko, B., Shiu, E. S. W. and Wei, L. (2010). Pricing maturity guarantee with dynamic withdrawal benefit. Insurance Math. Econom. 47, 216223.
[11] Kuznetsov, A. (2012). On the distribution of exponential functionals for Lévy processes with jumps of rational transform. Stoch. Process. Appl. 122, 654663.
[12] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
[13] Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, 2444.
[14] Kyprianou, A. E., Pardo, J. C. and Pérez, J. L. (2014). Occupation times of refracted Lévy processes. J. Theoret. Prob. 27, 12921315.
[15] Lee, H. (2003). Pricing equity-indexed annuities with path-dependent options. Insurance Math. Econom. 33, 677690.
[16] Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Prob. 45, 118134.
[17] MacKay, A., Augustyniak, M., Bernard, C. and Hardy, M. R. (2017). Risk management of policyholder behavior in equity-linked life insurance. J. Risk Insurance 84, 661690.
[18] Ng, A. C.-Y. and Li, J. S.-H. (2011). Valuing variable annuity guarantees with the multivariate Esscher transform. Insurance Math. Econom. 49, 393400.
[19] Pistorius, M. (2006). On maxima and ladder processes for a dense class of Lévy process. J. Appl. Prob. 43, 208220.
[20] Renaud, J.-F. (2014). On the time spent in the red by a refracted Lévy risk process. J. Appl. Prob. 51, 11711188.
[21] Situ, R. (2005). Theory of Stochastic Differential Equations with Jumps and Applications. Springer, New York.
[22] Wu, L. and Zhou, J. (2015). Occupation times of refracted Lévy processes with jumps having rational Laplace transform. Preprint. Available at https://arxiv.org/abs/1501.03363v3.
[23] Zhou, J. and Wu, L. (2015). Occupation times of refracted double exponential jump diffusion processes. Statist. Prob. Lett. 106, 218227.
[24] Zhou, J. and Wu, L. (2015). The time of deducting fees for variable annuities under the state-dependent fee structure. Insurance Math. Econom. 61, 125134.
[25] Zhou, J. and Wu, L. (2015). Valuing equity-linked death benefits with a threshold expense strategy. Insurance Math. Econom. 62, 7990.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

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