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Ruin probabilities with investments: smoothness, inegro-differential and ordinary differential equations, asymptotic behavior

Part of: Stochastic processes

Published online by Cambridge University Press:  24 June 2022

Yuri Kabanov  and
Nikita Pukhlyakov
Yuri Kabanov*
Affiliation:
Lomonosov Moscow State University and Université Bourgogne Franche-Comté
Nikita Pukhlyakov*
Affiliation:
Lomonosov Moscow State University
*
*Postal address: 16 Route de Gray, 25030 Besançon cedex, France.
*Postal address: 16 Route de Gray, 25030 Besançon cedex, France.
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Abstract

This study deals with the ruin problem when an insurance company having two business branches, life insurance and non-life insurance, invests its reserves in a risky asset with the price dynamics given by a geometric Brownian motion. We prove a result on the smoothness of the ruin probability as a function of the initial capital, and obtain for it an integro-differential equation understood in the classical sense. For the case of exponentially distributed jumps we show that the survival (as well as the ruin) probability is a solution of an ordinary differential equation of the fourth order. Asymptotic analysis of the latter leads to the conclusion that the ruin probability decays to zero in the same way as in the already studied cases of models with one-sided jumps.

Keywords

Ruin probabilitiesrisky investmentsactuarial models with investmentssmoothnessdifferential equations

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 556 - 570
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Albrecher, H., Gerber, H. and Yang, H. (2010). A direct approach to the discounted penalty function. N. Amer. Actuarial J. 14, 420–434.CrossRefGoogle Scholar
Belkina, T. (2014). Risky investments for insurers and sufficiency theorems for the survival probability. Markov Proc. Relat. Fields 20, 505–525.Google Scholar
Belkina, T. and Kabanov, Yu. (2015). Viscosity solutions of integro-differential equations for non-ruin probabilities. Theory Prob. Appl. 60, 802–810.Google Scholar
Belkina, T., Konyukhova, N. and Kurochkin, S. (2012). Singular boundary value problem for the integro-differential equation in an insurance model with stochastic premiums: Analysis and numerical solution. Comput. Math. Math. Phys. 52, 1384–1416.CrossRefGoogle Scholar
Feller, W. (1996). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York.Google Scholar
Frolova, A. G. (1994). Some mathematical models of risk theory. In Abstracts, All-Russian School-Colloq. Stoch. Meth. Geom. Anal., pp. 117–118.Google Scholar
Frolova, A., Kabanov, Yu. and Pergamenshchikov, S. (2002). In the insurance business risky investments are dangerous. Finance Stoch. 6, 227–235.CrossRefGoogle Scholar
Hsieh, P.-F. and Sibuya, Y. (1999). Basic Theory of Ordinary Differential Equations. Springer, Berlin.CrossRefGoogle Scholar
Kabanov, Yu. and Pergamenshchikov, S. (2016). In the insurance business risky investments are dangerous: The case of negative risk sums. Finance Stoch. 20, 355–379.CrossRefGoogle Scholar
Kabanov, Yu. and Pergamenshchikov, S. (2020). Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process. Finance Stoch. 24, 39–69.CrossRefGoogle Scholar
Pergamenshchikov, S. and Zeitouni, O. (2006). Ruin probability in the presence of risky investments. Stoch. Process. Appl. 116, 267–278. (Erratum: 119, 305–306.)CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Saxén, T. (1948). On the probability of ruin in the collective risk theory for insurance enterprises with only negative risk sums. Scand. Actuarial J., 199–228.CrossRefGoogle Scholar
Wang, G. and Wu, R. (2001). Distributions for the risk process with a stochastic return on investments. Stoch. Process. Appl. 95, 329–341.CrossRefGoogle Scholar