Characteristic function of non-Gaussian Ornstein-Uhlenbeck process

Question

I want to calculate the characteristic function $\phi(u)=\mathbb{E}\left[\exp(iuX_t)\right]$ of the following superposition of non-Gaussian Ornstein-Uhlenbeck process $$X_t=\int_0^t\left[e^{-a(t-s)}-e^{-b(t-s)}\right]\mathrm{d}Z_s,$$ where $a\neq b$ are two positive constants, $$Z_t=\sum_{i=1}^{N_t}J_i$$ is a compound Poisson process, $(N_t)_{t\geq 0}$ is a Poisson process with constant intensity $\lambda>0$, $J_i,\ i=1,2,\ldots,$ are mutually independent, exponentially distributed random variables with mean $\frac{1}{\delta}$, i.e., $$J_i\sim\mbox{Exp}(\delta).$$ I've tried to establish an SDE for $X_t$ or to rewrite it as sum of jumps since $Z$ is a pure jump process. It seems that neither approach works. Is there any standard routine to calculate characteristic function for such generalized OU process?