Finite and infinite time ruin probabilities in a stochastic economic environment. (English) Zbl 1047.60040

Summary: Let \((A_1,B_1,L_1), (A_2,B_2,L_2),\dots \) be a sequence of independent and identically distributed random vectors. For \(n\in \mathbb N\), denote \[ Y_n = B_1 + A_1B_1 +A_1A_2B_3+\cdots +A_1\cdots A_{n-1}B_n+A_1\cdots A_nL_n. \] For \(M>0\), define the time of ruin by \(T_M = \inf \,\{n\mid Y_n>M\}\) \((T_M=+\infty \), if \(Y_n\leq M\) for \(n=1,2,\dots )\). We are interested in the ruin probabilities for large \(M\). Our objective is to give reasons for the crude estimates \({\mathbf P}(T_M\leq x \log M)\approx M^{-R(x)}\) and \({\mathbf P}(T_M< \infty )\approx M^{-w}\) where \(x>0\) is fixed and \(R(x)\) and \(w\) are positive parameters. We also prove an asymptotic equivalence \({\mathbf P}(T_M<\infty )\sim CM^{-w}\) with a strictly positive constant \(C\). Similar results are obtained in an analogous continuous time model.


60G40 Stopping times; optimal stopping problems; gambling theory
60F10 Large deviations
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[1] de Acosta, A., Upper bounds for large deviations of dependent random vectors, Z. wahrsch. verw. geb., 69, 551-565, (1985) · Zbl 0547.60033
[2] Arfwedson, G., 1955. Research in collective risk theory. Part 2. Skand. Aktuarietidskr., 53-100. · Zbl 0067.12201
[3] von Bahr, B., 1974. Ruin probabilities expressed in terms of ladder height distributions. Scand. Actuarial J., 190-204. · Zbl 0321.62103
[4] Barndorff-Nielsen, O., Information and exponential families in statistical theory., (1978), Wiley Chichester, UK · Zbl 0387.62011
[5] Bertoin, J., Lévy processes., (1996), Cambridge University Press Cambridge
[6] Collamore, J.F., First passage times of general sequences of random vectors: a large deviations approach, Stochastic process. appl., 78, 97-130, (1998) · Zbl 0934.60025
[7] Cramér, H., 1955. Collective risk theory. Jubilee volume of Försäkringsbolaget Skandia, Stockholm.
[8] Daykin, C.D.; Pentikäinen, T.; Pesonen, M., Practical risk theory for actuaries., (1994), Chapman & Hall London · Zbl 1140.62345
[9] Dembo, A.; Zeitouni, O., Large deviations techniques and applications., (1993), Jones and Bartlett Publishers Boston · Zbl 0793.60030
[10] Embrects, P.; Goldie, C.M., Perpetuities and random equations., (), 75-86
[11] 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
[12] Goldie, C.M., Implicit renewal theory and tails of solutions of random equations, Ann. appl. probab., 1, 126-166, (1991) · Zbl 0724.60076
[13] Grey, D.R., Regular variation in the tail behaviour of solutions of random difference equations, Ann. appl. probab., 4, 169-183, (1994) · Zbl 0802.60057
[14] Martin-Löf, A., 1983. Entropy estimates for ruin probabilities. In: Gut, A., Holst, L. (Eds.), Probability and Mathematical Statistics. Department of Mathematics, Uppsala University, pp. 129-139.
[15] Martin-Löf, A., 1986. Entropy, a useful concept in risk theory. Scand. Actuar. J., 223-235. · Zbl 0649.62098
[16] Norberg, R., Ruin problems with assets and liabilities of diffusion type, Stochastic process. appl., 81, 255-269, (1999) · Zbl 0962.60075
[17] Norberg, R., 1999b. Lecture in the meeting on risk theory. Based on joint results with V. Kalashnikov, Oberwolfach.
[18] Nyrhinen, H., On the typical level crossing time and path, Stochastic process. appl., 58, 121-137, (1995) · Zbl 0829.60023
[19] Nyrhinen, H., Rough descriptions of ruin for a general class of surplus processes, Adv. appl. probab., 30, 1008-1026, (1998) · Zbl 0932.60046
[20] Nyrhinen, H., Large deviations for the time of ruin, J. appl. probab., 36, 733-746, (1999) · Zbl 0947.60048
[21] Nyrhinen, H., On the ruin probabilities in a general economic environment, Stochastic process. appl., 83, 319-330, (1999) · Zbl 0997.60041
[22] Paulsen, J., Risk theory in a stochastic economic environment, Stochastic process. appl., 46, 327-361, (1993) · Zbl 0777.62098
[23] 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
[24] Paulsen, J.; Gjessing, H.K., Ruin theory with stochastic return on investments, Adv. appl. probab., 29, 965-985, (1997) · Zbl 0892.90046
[25] Protter, P., Stochastic integration and differential equations., (1990), Springer Berlin
[26] Rockafellar, R.T., Convex analysis., (1970), Princeton University Press Princeton, NJ · Zbl 0229.90020
[27] Schnieper, R., 1983. Risk processes with stochastic discounting. Mitt. Verein. Schweiz. Vers. Math. 83 Heft 2, 203-218. · Zbl 0528.62088
[28] Segerdahl, C.-O., 1955. When does ruin occur in the collective theory of risk? Skand. Aktuarietidskr., 22-36. · Zbl 0067.12105
[29] Siegmund, D., 1975. The time until ruin in collective risk theory. Mitt. Verein. Schweiz. Vers. Math. 75 Heft 2, 157-166. · Zbl 0388.62091
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