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Finite time ruin probability of the compound renewal model with constant interest rate and weakly negatively dependent claims with heavy tails. (English) Zbl 1359.62469

The compound renewal risk model is considered in the paper. Let the individual claim sizes \(\{X_1,X_2, \dots\}\) form a sequence of weakly negatively dependent (w.n.d.) and identically distributed, nonnegative random variables. Let the inter-arrival times \(\{T_1,T_2, \dots\}\) form another sequence of w.n.d. and identically distributed random variables independent of \(\{X_1,X_2, \dots\}\). Suppose that the random number \(N_n\) of claims \(\{X_1^{(n)},X_2^{(n)}, \dots\}\) occur at time moment \(T_1+T_2+\dots +T_n\), where \(\{X_1^{(n)},X_2^{(n)}, \dots\}\) are independent copies of \(\{X_1,X_2, \dots\}\), and \(\{N_1,N_2, \dots\}\) are identically distributed integer-valued positive random variables. In addition, suppose that the random sequences \(\{X_1^{(n)},X_2^{(n)}, \dots\}_{n=1}^\infty\), \(\{N_1,N_2, \dots\}\) and \(\{T_1,T_2, \dots\}\) are mutually independent. The total surplus \(U_\delta(t)\) up to time \(t\) has the form \[ U_\delta(t)=xe^{\delta t}+\int_0^t e^{\delta(t-y)}C(dy)-\sum_{n=1}^{N(t)}S_{N_n}^{(n)}e^{\delta(t-T_1-T_2-\cdots-T_n)}, \] where \(x\geq 0\) is an initial surplus, \(\delta>0\) is a constant force of interest, \(N(t)\) is the renewal process generated by \(\{T_1,T_2, \dots\}\), \(S_{N_n}^{(n)}=\sum_{k=1}^{N_n}X_k^{(n)}\) and \(\{C(t),t\geq 0\}\) (with \(C(0)=0\)) is a non-decreasing, right continuous stochastic process representing the total amount of premiums accumulated up to time \(t\).
In the paper, the asymptotic formulas are derived for the probability of ruin \[ \psi_\delta(x,T)=\mathbb{P}\big(U_\delta(t)<0 \text{ for some } t\in [0,T]\big) \] within a finite time \(T>0\) as \(x\) tends to infinity.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62E10 Characterization and structure theory of statistical distributions
91B30 Risk theory, insurance (MSC2010)
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