Solution
= Solution
Put $q_x=-q_{x,x}=\sum_{y\ne x}q_{x,y}$. The <jump chain> records the successive states visited by $X$ and has transition probabilities
$$
P(x,y)=\frac{q_{x,y}}{q_x},\qquad y\ne x.
$$
By the exponential holding-time construction, conditional on the current state $x$, the destination of the next jump has this distribution independently of the earlier path. Hence the jump chain has the <Markov property>.