Probability Density Function of Compound Poisson Process

Question

I am trying to determine if it is possible to compute the probability density function (PDF) of a compound Poisson process $Y(t) = \sum_{i=0}^{N(t)} X_i$, where $N(t)$ is governed by a Poisson process with constant rate $\lambda$ and $X_i \sim \mathcal{N}(0,\sigma^2)$ are i.i.d. Gaussian random variables. To this end, we can compute the characteristic function (CF) of this compound Poisson process as $$ \phi_{Y_t}(s) \triangleq \mathbb{E}[e^{isY_t}] = \exp\left(\lambda t(\phi_{X}(s) - 1)\right). $$ The CF of a (zero-mean) normal random variable is $\phi_{X}(s) = \exp\left(-\frac{1}{2}\sigma^2 s^2\right)$, and thus plugging this in yields $$ \phi_{Y_t}(s) = \exp\left(\lambda t\left(\exp\left(-\frac{1}{2}\sigma^2 s^2\right)\right)\right). $$ To compute the PDF of this random variable, we apply the inverse Fourier transform to get $$ \psi_{Y_t}(x) = \frac{1}{2\pi}e^{-\lambda t}\int_{\mathbb{R}} e^{-isx}e^{\lambda t \exp\left(-\frac{1}{2}\sigma^2 s^2\right)} \ ds $$ This is the integral of an exponential of an exponential, which I have no idea how to solve, even if it's tractable or bounded. Does the PDF not exist? What can we say about this distribution?