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An insurance risk model with Parisian implementation delays. (English) Zbl 1319.60098
The authors study Lévy insurance risk processes where the surplus process is allowed to spend time under a pre-specified default level before ruin is recognized. This setup, called Parisian ruin, is motivated by the practical consideration that a regulator is unlikely to monitor the surplus level on a continuous basis, hence may not be immediately notified of a capital shortfall event. The implementation delays are stochastic and are specified to be of mixed Erlang nature, which is shown to improve the tractability of the Laplace transform of the ruin time, as compared to the approach in [I. Czarna and Z. Palmowski, J. Appl. Probab. 48, No. 4, 984–1002 (2011; Zbl 1232.60036)] where the delays are deterministic.
The authors present explicit formulas for the Laplace transform of the Parisian ruin time when the implementation delay is exponentially distributed or follows a mixed Erlang distribution. For the classical compound Poisson risk model, a numerical example is presented showing that if a deterministic delay time $$T$$ is approximated by a sequence of Erlang distributed implementation delays with mean $$T$$ and variance $$T^{2}/n$$, then these Parisian ruin probabilities converge to the Parisian ruin probability with a deterministic implementation delay; moreover, the Parisian ruin probabilities with Erlang delays are more conservative than their counterparts with deterministic delay.

##### MSC:
 60G51 Processes with independent increments; Lévy processes 91B30 Risk theory, insurance (MSC2010)
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