Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T05:49:53.735Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  30 November 2017

Jiang Zhou  and
Lan Wu
Jiang Zhou*
Affiliation:
Peking University
Lan Wu*
Affiliation:
Peking University
*
* Postal address: School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
* Postal address: School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Refracted Lévy processrational Laplace transformcontinuity theoremWiener–Hopf factorizationvariable annuitystate-dependent fee

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 91B30: Risk theory, insurance
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1167 - 1192
Copyright
Copyright © Applied Probability Trust 2017 

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] 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, 2062–2080. Google Scholar
[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, 79–111. CrossRefGoogle Scholar
[3] Bauer, D., Kling, A. and Russ, J. (2008). A universal pricing framework for guaranteed minimum benefits in variable annuities. ASTIN Bull. 38, 621–651. CrossRefGoogle Scholar
[4] Bernard, C., Hardy, M. and MacKay, A. (2014). State-dependent fees for variable annuity guarantees. ASTIN Bull. 44, 559–585. Google Scholar
[5] Cai, N. (2009). On first passage times of a hyper-exponential jump diffusion process. Operat. Res. Lett. 37, 127–134. CrossRefGoogle Scholar
[6] Cai, N., Chen, N. and Wan, X. (2009). Pricing double-barrier options under a flexible jump diffusion model. Operat. Res. Lett. 37, 163–167. Google Scholar
[7] Delong, Ł. (2014). Pricing and hedging of variable annuities with state-dependent fees. Insurance Math. Econom. 58, 24–33. Google Scholar
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York. Google Scholar
[9] Gerber, H. U. and Shiu, E. S. W. (1994). Option pricing by Esscher transforms. Trans. Soc. Actuaries 46, 99–140. Google Scholar
[10] Ko, B., Shiu, E. S. W. and Wei, L. (2010). Pricing maturity guarantee with dynamic withdrawal benefit. Insurance Math. Econom. 47, 216–223. Google Scholar
[11] Kuznetsov, A. (2012). On the distribution of exponential functionals for Lévy processes with jumps of rational transform. Stoch. Process. Appl. 122, 654–663. Google Scholar
[12] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin. Google Scholar
[13] Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, 24–44. Google Scholar
[14] Kyprianou, A. E., Pardo, J. C. and Pérez, J. L. (2014). Occupation times of refracted Lévy processes. J. Theoret. Prob. 27, 1292–1315. CrossRefGoogle Scholar
[15] Lee, H. (2003). Pricing equity-indexed annuities with path-dependent options. Insurance Math. Econom. 33, 677–690. CrossRefGoogle Scholar
[16] Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Prob. 45, 118–134. Google Scholar
[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, 661–690. CrossRefGoogle Scholar
[18] Ng, A. C.-Y. and Li, J. S.-H. (2011). Valuing variable annuity guarantees with the multivariate Esscher transform. Insurance Math. Econom. 49, 393–400. Google Scholar
[19] Pistorius, M. (2006). On maxima and ladder processes for a dense class of Lévy process. J. Appl. Prob. 43, 208–220. Google Scholar
[20] Renaud, J.-F. (2014). On the time spent in the red by a refracted Lévy risk process. J. Appl. Prob. 51, 1171–1188. CrossRefGoogle Scholar
[21] Situ, R. (2005). Theory of Stochastic Differential Equations with Jumps and Applications. Springer, New York. Google Scholar
[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. Google Scholar
[23] Zhou, J. and Wu, L. (2015). Occupation times of refracted double exponential jump diffusion processes. Statist. Prob. Lett. 106, 218–227. Google Scholar
[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, 125–134. CrossRefGoogle Scholar
[25] Zhou, J. and Wu, L. (2015). Valuing equity-linked death benefits with a threshold expense strategy. Insurance Math. Econom. 62, 79–90. Google Scholar