Abstract We consider a Lévy process reflected at the origin with additional independent and identically distributed 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 Lévy process is spectrally positive, and, finally, we specialize further to the two cases where the Lévy process is a Brownian motion and a compound Poisson process with exponential jumps minus a linear slope.
Boxma et al. (2026) studied this question.