It is well known, that if the domain of the mgf $M:=E[e^{uX}]$ of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments.
Consider the cumulant generating function $k:=\log E[e^{uX}]$ with domain $D_t$. Of course $0 \in D_t$. Assume moreover $D_t$ is an intervall. Can we already say, that the cumulant transform completely determines its distribution?
According to a statement of Jacod, Shiryaev Limit theorems for stochastic processes p. 612 mentoined in 2.3 Proposition, this should hold
We may have a process with domain $[0,\infty)$, which doesn't contain an interval around zero.
If not, with the assumptions of $D_t$ above, can we say, that if all moments of $X$ exists, then $k$ determines the distribution of $X$?