Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
권호 표지
DOI QR코드

DOI QR Code

Moments of the ruin time and the total amount of claims until ruin in a diffusion risk process

  • Kim, Jihoon (Department of Statistics, University of Seoul) ;
  • Ahn, Soohan (Department of Statistics, University of Seoul)
  • Received : 2015.12.31
  • Accepted : 2016.01.12
  • Published : 2016.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider a diffusion risk process, in which, its surplus process behaves like a Brownian motion in-between adjacent epochs of claims. We assume that the claims occur following a Poisson process and their sizes are independent and exponentially distributed with the same intensity. Our main goal is to derive the exact formula of the joint moment generating function of the ruin time and the total amount of aggregated claim sizes until ruin in the diffusion risk process. We also provide a method for computing the related first and second moments using the joint moment generating function and the augmented matrix exponential function.

Keywords

Acknowledgement

Supported by : University of Seoul

References

  1. Ahn, S. (2015). Total shift during the first passages of Markov modulated Brownian motion: Formulas driven by the minimal solution matrix of a Riccati equation. Accepted in Stochastic Models.
  2. Ahn, S., Badescu, A. L., Cheung, E. C. K. and Kim, J. (2016). Insurance risk models with marked Poisson arrivals. In preparation.
  3. Asmussen, S. (2003). Applied Probability and Queues, Second Edition, Springer Verlag.
  4. Chi, Y. (2010). Analysis of the expected discounted penalty function for a general jump-diffusion risk process and applications in finance. Insurance: Mathematics and Economics, 46, 385-396. https://doi.org/10.1016/j.insmatheco.2009.12.004
  5. Fung, T. C. (2004). Computation of the matrix exponential and its derivatives by scaling and squaring. International Journal for Numerical Methods in Engineering, 59, 1273-1286. https://doi.org/10.1002/nme.909
  6. Gerber, H. U. and Shiu, E. S. W. (1997). The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 21, 129-137. https://doi.org/10.1016/S0167-6687(97)00027-9
  7. Irving, R. (2013). Beyond the quadratic formula, The Mathematical Association of America.
  8. Lee, E. Y. (2014). Stationary analysis of the surplus process in a risk model with investments. Journal of the Korean Data & Information Science Society, 25, 915-920. https://doi.org/10.7465/jkdi.2014.25.4.915
  9. Won, H. J., Choi, S. K. and Lee, E. Y. (2013). Ruin probabilities in a risk process perturbed by diffusion with two types of claims. Journal of the Korean Data & Information Science Society, 24, 1-12. https://doi.org/10.7465/jkdi.2013.24.1.1
  10. Yang, H. and Zhang, C. (2005). Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 37, 615-634. https://doi.org/10.1016/j.insmatheco.2005.06.009
  11. Zhang, C. and Wang, G. (2003). The joint density function of three characteristics on jump-diffusion risk process. Insurance: Mathematics and Economics, 32, 445-455. https://doi.org/10.1016/S0167-6687(03)00133-1