= Solution
Conditional on $Y_m=i$, the <holding time> $V_m=U_m/g(i)$ has an <exponential distribution> of rate $g(i)$, and after that holding time the next state is $j$ with <transition probability> $p_{ij}$. The <memoryless property> of exponential random variables and the <Markov property> of $(Y_m)$ therefore make $(X_t)$ a <continuous-time Markov chain>.
Moreover $g(i)<1/\varepsilon$, so $V_m\geq\varepsilon U_m$. The <strong law of large numbers> gives $\sum_mU_m=\infty$ almost surely, hence $J_m\to\infty$ and the process is <nonexplosion under uniformly bounded jump rates>[nonexplosive]. Its <Q-matrix> is
$$
\boxed{q_{ij}=g(i)p_{ij}\quad(i\ne j),\qquad q_{ii}=-g(i).}
$$
Equivalently, if $D_g$ is the <diagonal matrix> with entries $g(i)$, then
$$
\boxed{Q=D_g(P-I).}
$$
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