Deficit distributions at ruin in a regime-switching Sparre Andersen model

In this paper, we investigate deficit distributions at ruin in a regime-switching Sparre Andersen model. A Markov chain is assumed to switch the amount and/or respective wait time distributions of claims while the insurer can adjust the premiums in response. Special attention is paid to an operator L generated by the risk process. We show that the deficit distributions at ruin during n periods, given the state of the Markov chain at time zero, form a vector of functions, which is the n-th iteration of L on the vector of functions being identically equal to zero. Moreover, in the case of infinite horizon, the deficit distributions at ruin are shown to be a fixed point of L. Upper bounds for the vector of deficit distributions at ruin are also proven.