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Saddlepoint approximations to the distribution of the total claim amount in some recent risk models. (English) Zbl 0780.62081

In the framework of risk theory the total claim amount is usually described by means of the formula: \(X(t)= \sum^{N(t)}_{i=1} X_ i\), where \(N(t)\) represents the number of claims, \(N(t)\sim\text{Poisson} (\beta t)\), and the claims \(X_ i\) are independent and identically distributed and also independent of \(N(t)\). A well-known approximation for the total claim distribution is the saddlepoint approximation or Esscher approximation. The purpose of the paper is the extension of the above mentioned approximations to other risk models.
At first, the author extends the saddlepoint approximation for \(P(X(t)>x)\) in the case of a claim process of the form: \[ X(t)= U(t)+ \sum^ d_{k=1} \sum^{N(t;k)}_{i=1} X_{k_ i}\quad\text{(model of K. K. Aase)}, \] where \(U(t)\) is a Gaussian random variable related to fluctuations in the premium, and the last term on the right hand side is a sum of \(d\) ordinary Poisson sums. Then, using G. E. Willmot’s results in ibid. 1989, No. 1, 1-12 (1989; Zbl 0679.62094), he investigates the classical model under inflationary conditions, and finally the Markov modulated risk process, which was proposed by S. Asmussen [see ibid. 1989, No. 2, 69-100 (1989; Zbl 0684.62073)] is studied.
The results concerning the mentioned models are stated by assuming that the claim amount distribution has a density with respect to the Lebesgue measure; in the final part of the paper the case where the claim distribution is a lattice distribution is discussed.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62E20 Asymptotic distribution theory in statistics
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[1] Aase K. K., Scand. Actuarial J. pp 65– (1985)
[2] Asmussen S., Scand. J. Statist. 16 pp 319–
[3] Asmussen S., Scand. Actuarial J. pp 69–
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[6] DOI: 10.2307/1427123 · Zbl 0576.62098 · doi:10.2307/1427123
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[8] Esscher F., Skand. Akt. Tidskr. pp 78– (1963)
[9] DOI: 10.2307/1427038 · Zbl 0656.60043 · doi:10.2307/1427038
[10] Jensen J. L., J. Roy. Statist. Soc. B. 53 pp 157– (1991)
[11] Willmot G. E., Scand. Actuarial J. pp 1– (1989)
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