zbMATH — the first resource for mathematics

On the absolute ruin in a map risk model with debit interest. (English) Zbl 1229.91171
A Markov-additive risk model \(\{U_t,J_t\}\) is considered, where \(U_t\) is the surplus and \(J_t\) is the state of the Markov process. If the surplus becomes negative, interest at rate \(r\) has to be paid for the deficit. The time of absolute ruin \(T\) is the first time where the payments for interest are larger than the premium income. The quantity of interest is the discounted penalty function \[ \Phi_{i j}(u) = E_{(u,i)} [e^{-\delta T} w(U_{T-}-c/r, c/r- U_T) I_{J_T = j}]\;, \] where \(c\) is the premium rate, \(w\) is a bounded measurable function, and \(I\) is the indicator function. The usual integro-differential equations are proved. In a complicated way, the boundary condition \(\Phi_{i j}(-c/r)\) is found. It would have been simpler to observe that \(\Phi\) is continuous in \(-c/r\) and that starting in \(u = -c/r\) means that absolute ruin occurs at the first claim time with \(U_{T-} = -c/r\).
Heavy-tailed claim sizes are then considered. Four classes are introduced: Subexponential and long-tailed distributions as well subexponential and long-tailed density functions. The asymptotic behaviour of \(\Phi\) for the case of subexponential distributions and subexponential densities is calculated.
Unfortunately, the paper contains some errors. For example, in the proof of Lemma 4 it is claimed that \(F\) long-tailed implies that also the density \(f\) of \(F\) is long-tailed. The following simple example gives a long-tailed distribution with an infinitely often differentiable density. Let \(h(x) = C \exp\{-1/(1-x^2)\}I_{|x| < 1}\) with \(C\) chosen such that \(\int_{-1}^1 h(x) \text{d} x = 1\). Then \[ f(x) = \text{\({1\over 2}\)} e^{-x} + \sum_{n=1}^\infty n h(2 n^2(n+1)(x-n^2))\;. \] Since \(e^x\) is not heavy-tailed, it does not matter asymptotically. The weight close to \(n^2\) is approximately \(1\over 2n(n+1)\). Thus the tail of \(F\) is \[ \sum_{n=x}^\infty {1\over 2n(n+1)} \sim \int_x^\infty {1\over 2y(y+1)} \text{d}y = {1\over 2} \log {x+1 \over x} \sim {1\over 2 x}\;. \] This proves that \(F\) is long-tailed. But for all \(y \neq 0\) \[ \limsup_{x \to \infty} {f(x+y) \over f(x)} = \limsup_{x \to \infty} {f(x)\over f(x+y)} = \limsup_{x \to \infty} f(x) = \infty\;. \]

91B30 Risk theory, insurance (MSC2010)
60J28 Applications of continuous-time Markov processes on discrete state spaces
91B70 Stochastic models in economics
Full Text: DOI
[1] Ahn, S. and Badescu, A. L. (2007). On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals. Insurance Math. Econom. 41 , 234-249. · Zbl 1193.60103 · doi:10.1016/j.insmatheco.2006.10.017
[2] Asmussen, S. (2000). Ruin Probabilities . World Scientific, River Edge, NJ. · Zbl 0960.60003
[3] Asmussen, S. (2003). Applied Probability and Queues , 2nd edn. Springer, New York. · Zbl 1029.60001
[4] Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behavior. J. Theoret. Prob. 16 , 489-518. · Zbl 1033.60053 · doi:10.1023/A:1023535030388
[5] Asmussen, S., Fløe Henriksen, L. and Klüppelberg, C. (1994). Large claims approximations for risk processes in a Markovian environment. Stoch. Process. Appl. 54 , 29-43. · Zbl 0814.60067 · doi:10.1016/0304-4149(93)00003-X
[6] Breuer, L. (2008). First passage times for Markov additive processes with positive jumps of phase type. J. Appl. Prob. 45 , 779-799. · Zbl 1156.60059 · doi:10.1239/jap/1222441829
[7] Badescu A. L. et al. (2005a). The joint density of the surplus prior to ruin and the deficit at ruin for a correlated risk process. Scand. Actuarial J. 2005 , 433-446. · Zbl 1143.91025 · doi:10.1080/03461230510009835
[8] Badescu A. L. et al. (2005b). Risk processes analyzed as fluid queues. Scand. Actuarial J. 2005 , 127-141. · Zbl 1092.91037 · doi:10.1080/03461230410000565
[9] Cai, J. (2007). On the time value of absolute ruin with debit interest. Adv. Appl. Prob. 39 , 343-359. · Zbl 1141.91023 · doi:10.1239/aap/1183667614 · euclid:aap/1183667614
[10] Cheung, E. C. K. and Landriault, D. (2010). A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model. Insurance Math. Econom. 46 , 127-134. · Zbl 1231.91156 · doi:10.1016/j.insmatheco.2009.07.009
[11] Dassios, A. and Embrechts, P. (1989). Martingales and insurance risk. Commun. Statist. Stoch. Models 5 , 181-217. · Zbl 0676.62083 · doi:10.1080/15326348908807105
[12] Embrechts, P. and Schmidli, H. (1994). Ruin estimation for a general insurance model. Adv. Appl. Prob. 26 , 404-422. JSTOR: · Zbl 0811.62096 · doi:10.2307/1427443 · links.jstor.org
[13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events . Springer, Berlin. · Zbl 0873.62116
[14] Gerber, H. U. 1971. Der Einfluss von Zins auf die Ruinwahrscheinlichkeit. Bull. Swiss Assoc. Actuaries 71 , 63-70. · Zbl 0217.26804
[15] Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2 , 48-78. · Zbl 1081.60550
[16] Gerber, H. U. and Yang, H. (2007). Absolute ruin probabilities in a jump diffusion risk model with investment. N. Amer. Actuarial J. 11 , 159-169.
[17] Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25 , 132-141. JSTOR: · Zbl 0651.60020 · doi:10.2307/3214240 · links.jstor.org
[18] Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82 , 259-269. · Zbl 0687.60017 · doi:10.1007/BF00354763
[19] Konstantinides, D. G., Ng, K. W. and Tang, Q. (2010). The probabilities of absolute ruin in the renewal risk model with constant force of interest. J. Appl. Prob. 47 , 323-334. · Zbl 1194.91094 · doi:10.1239/jap/1276784894
[20] Linz, P. (1985). Analytical and Numerical Methods for Volterra Equations (SIAM Stud. Appl. Math. 7 ). Society for Industrial and Applied Mathematics, Philadelphia, PA. · Zbl 0566.65094
[21] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance . John Wiley, Chichester. · Zbl 0940.60005
[22] Tang, Q. and Wei, L. (2010). Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence. Insurance Math. Econom. 46 , 19-31. · Zbl 1231.91243 · doi:10.1016/j.insmatheco.2009.08.007
[23] Yin, C. and Wang, C. (2010). The perturbed compound Poisson risk process with investment and debit interest. Methodology Comput. Appl. Prob. 12 , 391-413. · Zbl 1231.91255 · doi:10.1007/s11009-008-9109-z
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.