On the first passage times for Markov processes with monotone convex transition kernels (Q1899266)

For the Markov process \(Z_t\), \(t \geq 0\), with partially ordered Polish state space, the random time \[ T(x,y) = \inf \{t \geq 0 : Z_t > x \text{ or } Z_t - Z_{t-} > y\} \] is considered. It is shown that under certain conditions the time \(T\) possesses the NBU, IFRA or IFR properties. The main of the conditions is the monotonicity and convexity of the corresponding transition kernel. The obtained results are applied to the pure jump shock model.