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Ruin problems with stochastic premium stochastic return on investments. (English) Zbl 1148.60067
Summary: Ruin problems in the risk model with stochastic premium incomes and stochastic return on investments are studied. The logarithm of the asset price process is assumed to be a Lévy process. An exact expression for expected discounted penalty function is established. Lower bounds and two kinds of upper bounds for expected discounted penalty function are obtained by the inductive method and martingale approach. Integro-differential equations for the expected discounted penalty function are obtained when the Lévy process is a Brownian motion with positive drift and a compound Poisson process, respectively. Some analytical examples and numerical examples are given to illustrate the upper bounds and the applications of the integro-differential equations in this paper.

MSC:
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60G44 Martingales with continuous parameter
60J65 Brownian motion
60K05 Renewal theory
91B28 Finance etc. (MSC2000)
91B30 Risk theory, insurance (MSC2010)
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