On reflected L\'evy processes with collapse

We consider a L\'evy process reflected at the origin with additional i.i.d. collapses that occur at Poisson epochs, where a collapse is a jump downward to a state which is a random fraction of the state just before the jump. We first study the general case, then specialize to the case where the L\'evy process is spectrally positive and finally we specialize further to the two cases where the L\'evy process is a Brownian motion and a compound Poisson process with exponential jumps minus a linear slope.