Stochastic control for spectrally negative Lévy processes
نویسنده
چکیده
Three optimal dividend models are considered for which the underlying risk process is a spectrally negative Lévy process. The first one concerns the classical dividends problem of de Finetti for which we give sufficient conditions under which the optimal strategy is of barrier type. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal. The second one is a generalized version of the classical optimal dividends problem of de Finetti in which the objective function has an extra term which takes account of the ruin time of the risk process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. The third is an impulse control version of de Finetti’s dividends problem. Here we show that when the Lévy measure has a log-convex density, then an optimal strategy is given by paying out a dividend in such a way that the reserves are reduced to a certain level c1 whenever they are above another level c2. Also a method to numerically find the optimal values of c1 and c2 is presented. Finally, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. We consider in particular the case that X is spectrally negative and besides showing the existence of refracted Lévy processes, we establish a suite of identities for the case of one and two sided exit problems. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.
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تاریخ انتشار 2008