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Refracted Lévy processes. (English) Zbl 1201.60042
The paper is devoted to the study of the dynamics of a refracted Lévy process, which is one-dimensional Lévy process, perturbed in a special way: a linear drift is subtracted from its increments whenever it exceeds a predetermined positive level. More formally, the authors consider a solution to the stochastic differential equation \[ U_t = X_t - \delta\int_0^t \mathbf{1}_{\{U_s>b >}ds,\quad t\geq 0, \] where \(X_t\) is a Lévy process, \(b\) is a predetermined level. The solution of this equation may be thought of as the aggregate of the insurance risk process, when the dividends are paid out at a rate \(\delta\) whenever it exceed the level \(b\).
The authors show that refracted Lévy processes exist as strong solutions to the above stochastic differential equation whenever \(X\) is a spectrally negative Lévy process. Further they investigate the dynamics of such processes and establish a suite of identities, written in terms of scale functions, related to one- and two-sided exit problems. Finally they cite the relevance of such identities in context of a number of applications of spectrally negative Lévy processes within the context of ruin probabilities.

MSC:
60G51 Processes with independent increments; Lévy processes
91B70 Stochastic models in economics
93E20 Optimal stochastic control
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