Kac-L\'evy processes

Markov-modulated L'evy processes with two different regimes of restarting are studied. These regimes correspond to the completely renewed process and to the process of Markov modulation, accompanied by jumps. We give explicit expressions for the L'evy-Khintchine exponent in the case of a two-state underlying Markov chain. For the renewal case, the limit distributions (as $t\to \mathrm{\infty }$) are obtained. In the case of processes with jumps, we present some results for the exponential functional.