Stopped processes of temporally homogeneous Markov jump processes (Q1200713)

The authors prove that a homogeneous Markov (pure) jump process stopped at a hitting time is still a homogeneous Markov jump process and conversely, if a homogeneous Markov jump process \(X\) stopped at a stopping time \(T\) is also a homogeneous Markov jump process, then \(T\wedge S\) is a hitting time, where \(S=\inf\{t\geq 0\): \(q(X_ t)=0\}\), \(q(x)=\lim_{t\downarrow 0}{1\over t}(1-P(t,x,\{x\})\).