Martingales and first passage times for Ornstein–Uhlenbeck processes with a jump component

Using martingale technique, we show that a distribution of the first-passage time over a level for the Ornstein–Uhlenbeck process with jumps is exponentially bounded. In the case of absence of positive jumps, the Laplace transform for this passage time is found. Further, the maximal inequalities are also given when the marginal distribution is stable.