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Ruin probabilities and decompositions for general perturbed risk processes. (English) Zbl 1061.60075
Let $$C=(C(t),t\geq0)$$ be a subordinator without a drift and with finite expectation and let the risk process $$R=(R(t),t\geq0)$$ with premium rate $$c>0$$ be defined by $$R(t)=ct-C(t)$$. The authors consider a perturbed risk process $$X(t)=R(t)+Z(t)$$ for $$t\geq0$$, where the perturbation $$Z=(Z(t),t\geq0)$$ is a spectrally negative, mean zero, Lévy process, which is independent of the process $$C$$. This setting includes the Brownian perturbation and also the perturbation by $$\alpha$$-stable spectrally negative Lévy process for $$\alpha\in(1,2)$$. The aim of the paper is to derive the Pollaczek-Khinchin formula for the survival probability $$\theta(x):=P(X(t)\geq -x$$, $$\forall\,t\geq0)$$ and to give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 60G51 Processes with independent increments; Lévy processes 60J75 Jump processes (MSC2010) 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.) 91B30 Risk theory, insurance (MSC2010)
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