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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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##### References:
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