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Two-sided bounds for the finite time probability of ruin. (English) Zbl 0958.91030
This paper deals with the following risk process of an insurance company \[ R_{t}=h(t)-S_{t}, \] where \(h(t)\) is a nonnegative increasing real function defined on \(\mathbb{R}_{+}\) and such that \(\lim_{t\to+\infty}=+\infty,\) representing the premium income; \(S_{t}=\sum_{i=1}^{N_{t}}W_{i}\) is the aggregate claims amount at time \(t,\) where \(N_{t}=\#\{i: \tau_{1}+\dots+\tau_{i}\leq t\},\) \(\#\) is the number of elements in the set \(\{\cdot\},\) and \(\tau_{1},\dots, \tau_{i},\dots\) are independent exponentially distributed r.v.s; \(W_{1}, W_{2},\dots\) are dependent integer valued r.v.s independent of \(N_{t}\) which denote the severities of the successive claims. Explicit two-sided bounds are derived for the probability of ruin of an insurance company. It is shown that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula obtained recently by Ph. Picard and C. Lefevre [Scand. Actuarial J. 1997, No. 1, 58-69, (1997; Zbl 0926.62103)] in terms of the generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. Relations of the survival probability with the multivariate \(B\)-splines are considered.

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
41A15 Spline approximation
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