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We study the the stochastic properties of the area under some function of thedifference between (i) a spectrally positive L\'evy process $W_t^x$ that jumpsto a level $x>0$ whenever it hits zero, and (ii) its reflected version $W_t$.Remarkably, even though the analysis of each of these processes is challenging,we succeed in attaining explicit expressions for their difference. The mainresult concerns the Laplace-Stieltjes transform of the integral $A_x$ of (afunction of) the distance between $W_t^x$ and $W_t$ until $W_t^x$ 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 paper with aninventory problem for which our results are particularly useful.

On the area between a L\'evy process with secondary jump inputs and its reflected version