Solution
= Solution
The total jump rate is $\lambda_0$ at zero and $\lambda_i(1+\rho_i)$ at $i>0$. Hence the <jump chain> has transition probabilities
$$
P_{0,1}=1,
$$
and, for $i>0$,
$$
\boxed{P_{i,i+1}=\frac1{1+\rho_i},
\qquad
P_{i,0}=\frac{\rho_i}{1+\rho_i}.}
$$
All other entries vanish.