Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 28K / a / 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 probabilitiesBy 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.