Theory of Probability and Mathematical Statistics
Analytic properties of infinite-horizon survival probability in a risk model with additional funds
Yu. S. Mishura, O. Yu. Ragulina, O. M. Stroev
Download PDF
Abstract: We consider a generalization of the classical risk model where an insurance company gets additional funds whenever a claim arrives. We investigate the properties of continuity and differentiability of the infinite-horizon survival probability and derive an integro-differential equation. We find a closed form solution of this equation in the case where the claim sizes and additional funds are exponentially distributed.
Keywords: Risk model, survival probability, continuity and differentiability, integro-differential equation
Bibliography: 1. T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels, Stochastic Processes for Insurance and Finance, John Wiley & Sons, Chichester, 1999.
2. S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000.
3. J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991.
4. M. M. Leonenko, Yu. S. Mishura, V. M. Parkhomenko, and M. I. Yadrenko, Probabilistic and Statistical Methods in Econometrics and Financial Mathematics, Informtekhnika'', Kyiv, 1995. (Ukrainian)
5. A. V. Boikov, The Cramér-Lundberg model with stochastic premium process, Teor. Veroyatnost. Primenen. 47 (2002), no. 3, 549-553; English transl. in Theory Probab. Appl. 47 (2002), no. 3, 489-493.
6. O. Yu. Ragulina, The differentiability of the survival probability for an insurance company acting in a risk model with constant interest rate, Appl. Statist. Actuar. Finance Math. (2010), no. 1-2, 82-116.
7. O. Yu. Ragulina, Estimates for and properties of the survival probability of an insurance company in the classical risk model investing the capital in a financial (B,S)-market, Visnyk Kyiv Nat. Taras Shevchenko Univ. Ser. Fiz. Mat. (2012), no. 2, 23-30. (Ukrainian)
8. O. Yu. Ragulina, On the survival probability for two risk models, Appl. Statist. Actuar. Finance Math. (2012), no. 1, 40-50. (Ukrainian)
9. B. V. Bondarev and E. Yu. Ragulina, On the finite-time nonruin probability of an insurance company with investments in the financial (B,S)-market, Cybernetics Systems Anal. 48 (2012), no. 5, 736-748.
10. A. V. Skorokhod, Random Processes with Independent Increments, Nauka'', Moscow, 1964; English transl. in Mathematics and its Applications (Soviet Series), vol. 47, Kluwer Academic Publishers Group, Dordrecht, 1991.
11. D. V. Gusak, The cumulant representation of the Lundberg root in the case of semicontinuous processes, Teor. Imovirnost. Matem. Statist. 82 (2010), 21-29; English transl in Theor. Probability and Math. Statist. 82 (2011), 1-10.
12. S. Karlin, A First Course in Stochastic Processes, Academic Press, New York and London, 1968.
