Changing domain of the CGF of a stochastic process

Question

Given a stochastic cadlag process $(X_t)_{t\geq 0}$. Define $A_{t}$ as the domain of the cumulant generating function by $f_t(s):=\log E(e^{sX_t})$, for which this expression is welldefined. I search for an example, where the domain $A_{t}$ changes in time. So for some $t_1\not= t_2$ that $A_{t_1}$ and $A_{t_{2}}$ don't coincide.