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Upper bounds for ruin probabilities in a new general risk model, by the martingales method

Authors
Journal
Journal of Computational and Applied Mathematics
0377-0427
Publisher
Elsevier
Publication Date
Volume
8
Issue
2
Identifiers
DOI: 10.1016/0771-050x(82)90065-1
Disciplines
  • Mathematics

Abstract

Abstract We consider a portfolio in an insurance business of stochastically variable size in time. The portfolio is composed of independent indentical contracts, each of fixed duration, say one year. The contract number process is a general random point process defined by a sequence of intensities. The premium income process is defined as follows: for each new contract joining the portfolio, the insurer receives the premium c(1 + η) at the very instant it joins the portfolio. Here c is the expectation of the claim amount for a fixed contract and η is the security loading. Finally, the insurer is also supposed to possess an initial risk reserve u. Essential differences of the involved stochastic processes, • - the generality of the involved stochastic processes, • - the discontinuous, more realistic, character of the premium income process. The tribute to pay for this generality is that it is an almost impossible task to calculate exact ruin probabilities. But we can obtain bounds for the ruin probability in infinite or finite time intervals. In order to do this, we replace the surplus process by a minimizing surplus process and to the latter we apply the martingale argument used by Gerber [4] in the extended form employed by De Vylder [2].

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