 Solution

The jump chain rises from $i$ with probability $(i+1)/(i+2)$ and resets to $0$ otherwise. The probability of reaching level $i$ in an excursion is $2/(i+1)$, so reset occurs almost surely but the expected excursion length diverges: recurrence is null. Holding means grow like $i+1$, so there is no explosion and the expected return time is infinite. Hence neither the chain nor its jump chain is positive recurrent, and the continuous-time chain has no invariant probability distribution.