On the ruin probabilities in a general economic environment. (English) Zbl 0997.60041

The paper analyzes the asymptotic behavior of ruin probabilities for insurance companies with large initial capital. Given that the capital \(U\) of the company evolves in discrete time according to the rule \(U_{n}=A_{n}^{-1}(U_{n-1}-B_{n})\), where \(A_{n}=1/(1+r_{n})\) is the inverse of the rate of return on investment, \(B_{n}\) is the insurance payout net of premia at date \(n\) (\(n=1,2,...\)), the initial value of the company is \(U_{0}=M\) and \(T_{M}=\inf\{ n\mid U_{n}<0 \}\) is the ruin time, the author lists the conditions under which \(\lim _{M\rightarrow \infty } \log\text{Pr}\{ T_{M}<\infty \} /\log M =-w\) for some positive \(w\). The first group of conditions that guarantee the above asymptotic result includes independence of the return rate and the net payment processes \(A\) and \(B\), asymptotic growth requirements on \(A_{n}\) plus certain uniform integrability requirements on \(B_{n}\), and asymptotic uniform lower bound requirements on \(B_{n}\). The second group of conditions, yielding a refinement of the general result, adds to the above named the i.i.d. property of sequences \(A\) and \(B\) as well as a restriction on the convolution powers of the distribution of \(\log A_{1}\).


60G40 Stopping times; optimal stopping problems; gambling theory
60F10 Large deviations
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[1] Asmussen, S., Bladt, M., 1996. Phase-type distributions and risk processes with state-dependent premiums. Scand. Actuarial J. 19-36. · Zbl 0876.62089
[2] Bühlmann, H., Experience rating and credibility, Astin bull., 4, 199-207, (1967)
[3] Daykin, C.D., Pentikäinen, T., Pesonen, M., 1994. Practical Risk Theory for Actuaries. Chapman and Hall, London. · Zbl 1140.62345
[4] Dembo, A., Zeitouni, O., 1993. Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston. · Zbl 0793.60030
[5] Gjessing, H.K.; Paulsen, J., Present value distributions with applications to ruin theory and stochastic equations, Stochastic process. appl., 71, 123-144, (1997) · Zbl 0943.60098
[6] Goldie, C.M., Implicit renewal theory and tails of solutions of random equations, Ann. appl. probab., 1, 126-166, (1991) · Zbl 0724.60076
[7] Iscoe, I.; Ney, P.; Nummelin, E., Large deviations of uniformly recurrent Markov additive processes, Adv. appl. math., 6, 373-412, (1985) · Zbl 0602.60034
[8] Martin-Löf, A., 1983. Entropy estimates for ruin probabilities. In: Gut, A., Holst, L. (Eds.), Probability and Mathematical Statistics. Dept. of Mathematics, Uppsala University, pp. 129-139.
[9] Martin-Löf, A., 1986. Entropy, a useful concept in risk theory. Scand. Actuarial J. 223-235. · Zbl 0649.62098
[10] Møller, C.M., Stochastic differential equations for ruin probabilities, J. appl. probab., 32, 74-89, (1995) · Zbl 0816.60043
[11] Nyrhinen, H., Rough limit results for level-crossing probabilities, J. appl. probab., 31, 373-382, (1994) · Zbl 0808.60042
[12] Nyrhinen, H., Rough descriptions of ruin for a general class of surplus processes, Adv. appl. probab., 30, 1008-1026, (1998) · Zbl 0932.60046
[13] Paulsen, J., Risk theory in a stochastic economic environment, Stochastic process. appl., 46, 327-361, (1993) · Zbl 0777.62098
[14] Paulsen, J., Sharp conditions for certain ruin in a risk process with stochastic return on investments, Stochastic process. appl., 75, 135-148, (1998) · Zbl 0932.60044
[15] Paulsen, J., Ruin theory with compounding assets — a survey, Insurance: math. and econom., 22, 3-16, (1998) · Zbl 0909.90115
[16] Paulsen, J.; Gjessing, H.K., Ruin theory with stochastic return on investments, Adv. appl. probab., 29, 965-985, (1997) · Zbl 0892.90046
[17] Rockafellar, R.T., 1970. Convex Analysis. Princeton University Press, Princeton, NJ. · Zbl 0193.18401
[18] Ruohonen, M., On the probability of ruin of risk processes approximated by a diffusion process, Scand. actuarial J., 113-120, (1980) · Zbl 0427.62075
[19] Schnieper, R., 1983. Risk processes with stochastic discounting. Mitt. Verein. Schweiz. Vers. Math. 83 Heft 2, 203-218. · Zbl 0528.62088
[20] Wilkie, A.D., A stochastic investment model for actuarial use, Trans. faculty actuaries, 39, 341-373, (1986)
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