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Media type:
E-Article
Title:
Ruin probabilities with investments: smoothness, inegro-differential and ordinary differential equations, asymptotic behavior
Contributor:
Kabanov, Yuri;
Pukhlyakov, Nikita
Published:
Cambridge University Press (CUP), 2022
Published in:
Journal of Applied Probability, 59 (2022) 2, Seite 556-570
Language:
English
DOI:
10.1017/jpr.2021.74
ISSN:
0021-9002;
1475-6072
Origination:
Footnote:
Description:
AbstractThis 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.