Moment boundedness for hybrid linear systems with state resetting

Consider the hybrid linear system with jumps

$$dX_t=AX_t+BdW_t+CdL_t+(X_0-X_{t-})dN_t,$$

where $W_t$ is a Wiener process, $N_t$ is a counting process, and $L_t$ satisfies a moment condition of order $Q$. Moment-boundedness claim. If $L_t$ satisfies the moment condition of order $Q$, then

$$\sup_{t \geq 0}\mathbb{E}[|X|_t^q] < \infty$$

holds for $1 \leq q \leq Q$.

This claim concerns uniform-in-time moment stability for a hybrid linear system whose state is reset at the jump times of a counting process. The supplied text does not indicate whether the claim is conjectural or has been established.