Article http://dx.doi.org/10.26855/jamc.2023.12.008

Joint Moments of Surplus Immediately Before Ruin and Deficit at Ruin in the Renewal Risk Model


Jiayu Wang1, Houchun Wang2,*

1School of Mathematics, Hefei University of Technology, Hefei, Anhui, China.

2School of Mathematics & Physics, Anhui Jianzhu University, Hefei, Anhui, China.

*Corresponding author: Houchun Wang

Published: January 24,2024


This paper studies the risk process where the individual claim amount density and the interclaim time density are assumed to be arbitrary. It uses the renewability of the risk process at the time of claims and the cumulative law of mathematical expectation to introduce a defective renewal equation satisfied by the joint moments of the surplus immediately before ruin and the deficit at ruin. From this equation, an explicit expression for the above joint moments (including the ruin probability) is obtained by the Laplace transform. Explicit expressions for the ruin probability are given for the case where the claim amount distribution is exponential, and numerical examples are given to analyze the effect of the relevant parameters on the ruin probability under the conditions that the interclaim times follow a Gamma distribution and a generalized Gamma distribution, respectively.


[1] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. 2th Edition, World Scientific Publishing Company Limited, Singapore. 

[2] Cheung, E.C.K. and Liu, H. (2023). Joint Moments of Discounted Claims and Discounted Perturbation until Ruin in the Compound Poisson Risk Model with Diffusion. Probability in the Engineering and Informational Sciences, 37, 387-417.

[3] Wang, H. (2013). On the Ruin Probability for a Generalized Erlang(2) Risk Model Involving Two Classes of Insurance Risks. Chinese Journal of Engineering Mathematics, 30, 661-672. 

[4] Li B., Willmot, G.E. and Wong, J.T.Y. (2018). A Temporal Approach to the Parisian Risk Model. Journal of Applied Probability, 55, 302-317.

[5] Landriault, D., Li, B., Shi, T., and Xu, D. (2019). On the Distribution of Classic and Some Exotic Ruin Times. Insurance: Mathematics and Economics, 89, 38-45.

[6] Landriault, D., Li, B., and Lkabous, M.A. (2021). On the Analysis of Deep Drawdowns for the Lévy Insurance Risk Model. Insurance: Mathematics and Economics, 100, 147-155.

[7] Gerber, H.U. and Shiu, E.S.W. (1998). On the Time Value of Ruin. North American Actuarial Journal, 2, 48-78.

[8] Wang, H. and Ling, N. (2017). On the Gerber-Shiu Function with Random Discount Rate. Communications in Statistics–Theory and Methods, 46, 210-220. 

[9] Cheung, E.C.K. and Zhang, Z. (2021). Simple Approximation for the Ruin Probability in Renewal Risk Model under Interest Force via Laguerre Series Expansion. Scandinavian Actuarial Journal, 9, 804-831.

[10] Adekambi, F. and Takouda, E. (2022). On the Discounted Penalty Function in a Perturbed Erlang Renewal Risk Model with Dependence. Methodology and Computing in Applied Probability, 24, 481-513.

[11] He, Y., Kawai, R., Shimizu, Y., and Yamazaki, K. (2023). The Gerber-Shiu Discounted Penalty Function: A Review from Practical Perspectives. Insurance: Mathematics and Economics, 109, 1-28.

How to cite this paper

Joint Moments of Surplus Immediately Before Ruin and Deficit at Ruin in the Renewal Risk Model

How to cite this paper: Jiayu Wang, Houchun Wang. (2023) Joint Moments of Surplus Immediately Before Ruin and Deficit at Ruin in the Renewal Risk ModelJournal of Applied Mathematics and Computation7(4), 483-489.

DOI: http://dx.doi.org/10.26855/jamc.2023.12.008