In this paper, a dividend optimization problem with a terminal value at creeping ruin for Lévy risk models has been investigated. We consider an insurance company whose surplus process evolves as a spectrally negative Lévy process with a Gaussian part and its objective function is given by cumulative discounted dividend payments and a terminal value at creeping ruin. In views of identities from fluctuation theory, under the restriction on the negative terminal value, we show that the threshold strategy turns out to be the optimal one with threshold level at zero over an admissible class with restricted dividend rates. Furthermore, some sufficient conditions for the positive one also have been given.

The core of the research is to provide the explicit expression for the expected net present values (NPVs) of the double barrier strategy for regular diffusions. Under the so-called bail-out setting, the value of the expected NPVs of an insurance company varies according to the choice of a pair of policies, which consist of dividend payments paid out and capital injections received. In the case of the double barrier strategy, the closed-form expected NPVs are given via the bivariate q-scale function. This is accomplished by making use of a perturbation technique in [CP14], which could lead to the linear equation system. The expression obtained here shall be conducive to addressing the associated dividends control problems.

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