The Gerber-Shiu Discounted Penalty Function for Risk Process with Double Markovian Environment

Abstract:

In this paper, we study the Gerber-Shiu discounted penalty function. We shall consider the case where the discount interest process and the occurrence of the claims are driven by two distinguished Markov process, respectively. Moreover, in this model we also consider the influence of a premium rate which varies with the level of free reserves. Using backward differential argument, we derive the integral equation satisfied by the expected discounted penalty function via differential argument when interest process in every state is perturbed by standard Wiener process and Poisson process. In some special cases, closed form expression for these quantities are obtained.

• Periodical: Advanced Materials Research (Volumes 108-111)
• Pages: 1103-1108
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.108-111.1103
• Online since: May 2010
• Authors: Wen Guang Yu
• Keywords: Expected Discounted Penalty Function, Markov Process, Poisson Process, Standard Wiener Process, Stochastic Interest

References

[1] J.M. Reinhard: On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. ASTIN Bull, Vol. 14 (1984), p.23.

DOI: 10.1017/s0515036100004785

[2] S. Asmussen: Risk theory in a Markovian environment, Scandinavian Actuarial Journal, Vol. 2 (1989), p.69.

[3] N. Bauerle: Some results about the expected ruin time in Markov-modulated risk models. Insurance: Mathematics and Economics, Vol. 18 (1996), p.119.

DOI: 10.1016/0167-6687(95)00034-8

[4] A. Ng and H. Yang: Lundberg-type bounds for the joint distribution of surplus immediately before and after ruin under a Markov-modulated risk model. Astin Bulletin, Vol. 35 (2005), p.351.

DOI: 10.2143/ast.35.2.2003457

[5] A. Ng and H. Yang: On the joint distribution of surplus prior and immediately after ruin under a Markovian regime switching model. Stochastic Processes and their Applications, Vol. 116 (2006), pp.244-266.

DOI: 10.1016/j.spa.2005.09.008

[6] H. Jasiulewicz: Probability of ruin with variable premium rate in a Markovian environment. Insurance: Mathematics and Economics, Vol. 29 (2001), p.291.

DOI: 10.1016/s0167-6687(01)00090-7

[7] R. Wu and L. Wei: The probability of ruin in a kind of Cox risk model with variable premium rate. Scandinavian Actuarial Journal, Vol. 2004 (2004) No. 2, p.121.

DOI: 10.1080/03461230310017216

[8] X. Zhao and J. Liu: A ruin problem about classical risk process under random interest force. Appl. Math. J. Chinese Univ. Ser. A, Vol. 20 (2005), p.313.

[9] W.G. Yu: A m-type risk model with markov-modulated premium rate. Journal of Applied Mathematics and Informatics, Vol. 27 (2009) No. (5-6), p.1033.

[10] J. Cai and D.C.M. Dickson: On the expected discounted penalty function at ruin of a surplus process with interest. Insurance: Mathematics and Economics, Vol. 30 (2002), p.389.

DOI: 10.1016/s0167-6687(02)00120-8

[11] L. Liu and S. Mao: The risk model of the expected discounted penalty function with constant interest force. Acta Mathematica Scientia, Vol. 26B(2006), p.509.

DOI: 10.1016/s0252-9602(06)60076-3