On a doubly reflected risk process with running maximum dependent reflecting barriers

This paper investigates a spectrally negative Lévy process that is reflected from above when it is about to up-cross a dynamic upper barrier and is reflected from below when it is about to down-cross a dynamic lower barrier. These reflecting barriers are determined by pre-specified functions of the maximum value in historical record of the underlying spectrally negative Lévy process. Applying the techniques recently developed in fluctuation and excursion theories, we obtain the expressions for the solutions to the expected total amount of discounted dividends and capital injections and the nth moment of the accumulated discounted dividends. Moreover, the joint Laplace transform of the accumulated non-discounted dividends and capital injections prior to some exiting time and the associated potential measure are derived. These results are expressed in terms of scale functions associated with the spectrally negative Lévy processes. Finally, some numerical examples are provided to illustrate the economic implications of model parameters.