Is every strictly convex smooth function on a half space a cumulant generating function for some random variable

Question

Fix $a \in \mathbb{R}$ and let $\Psi(a, \infty) \mapsto \mathbb{R}$ be strictly convex and infinitely differentiable. Does there exist a probability measure $P$ on $\mathbb{R}$ such that $\Psi$ is the CGF of $P$? That is, $\Psi(u) = \log \int e^{y \cdot u} P(dy)$.