On the area between a Lévy process with secondary jump inputs and its reflected version

We study the stochastic properties of the area under some function of the difference between (i) a spectrally positive Lévy process (Formula presented.) that jumps to a level x > 0 whenever it hits zero and (ii) its reflected version W_t. Remarkably, even though the analysis of each of these areas is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral A_x of (a function of) the distance between (Formula presented.) and W_t until (Formula presented.) hits zero. This result is extended in a number of directions, including the area between A_x and A_y and a Gaussian limit theorem. We conclude the article with an inventory problem for which our results are particularly useful.