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On the expectation of total discounted operating costs up to default and its applications. (English) Zbl 1173.91023
Within the analysis of the surplus process of an insurance company, the paper focuses on quantities connected to the time of ruin. In particular, the authors investigate the expectation of the total discounted claim costs up to the time of ruin (say \(C\)) and the expectation of the total discounted operating costs (say \(H\)) up to the time of default, which is the first passage time of the surplus process going below any given level. Such a level can be linked to the probability of default. The above two expectations generalize well-known risk parameters in the literature, as the Gerber-Shiu function.
After introducing the piecewise compound Poisson process, the authors deepen the relationship between \(C\) and \(H\) and derive an integro-differential equation involving \(H\). Then a general solution to \(H\) in the classical compound Poisson process is given.
Successively several applications are presented by means of the accumulated utility up to ruin, the total discounted claim costs up to ruin, the Gerber-Shiu function with two-sided jumps and the pricing formula for a perpetual American put option with two-sided jumps.

91B30 Risk theory, insurance (MSC2010)
91B70 Stochastic models in economics
91G20 Derivative securities (option pricing, hedging, etc.)
Full Text: DOI
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