Ruin theory for a Markov regime-switching model under a threshold dividend strategy

In this paper, we study a Markov regime-switching risk model where dividends are paid out according to a certain threshold strategy depending
on the underlying Markovian environment process. We are interested in these quantities: ruin probabilities, deficit at ruin and expected ruin time.
To study them, we introduce functions involving the deficit at ruin and the indicator of the event that ruin occurs.We show that the above functions
and the expectations of the time to ruin as functions of the initial capital satisfy systems of integro-differential equations. Closed form solutions
are derived when the underlying Markovian environment process has only two states and the claim size distributions are exponential.