Let $Y_t$ be a continuous time stochastic process with characteristic function
$$ \varphi_{Y_t}\left(\omega\right)=\mathbb{E}\left[e^{i\,\omega\,Y_t}\right]. $$
For all $t$ let $\tau_t$ be a stopping time with density $f_t\left(z\right)$. I need to calculate the characteristic function of $Y_{\tau_t}$, that is of the process $Y$ randomly sampled at $\tau_t$. My guess is
$$ \varphi_{Y_{\tau_t}}=\mathbb{E}\left[e^{i\,\omega\,Y_{\tau_t}}\right] = \int_{\mathbb{R}}\varphi_{Y_z}\left(\omega\right)\,f_t\left(z\right)\,dz $$
am I right? It seems quite intuitive to me but I miss a formal justification.