We know that if $X = [X_t , t \in \mathbb{Z}]$ is a Levy process, then any marginal distribution has the charasteristic function - ch. f. of $X_t$ - given by $\varphi_t(r)=e^{t \phi(r)}$ whith: $$\phi(r) = iar - \frac{\sigma r^2}{2}+\int_{\mathbb{R}}\big(e^{irx} - 1 - irx \big)\frac{1}{x^2} d\kappa, \quad r \in \mathbb{R}$$ where $\kappa$ is a measure such that $\kappa(\mathbb{R})< \infty$.
I want to know if some reciprocal is true. More especifically, I have a random process $X = [X_t , t \in \mathbb{Z}]$ such that any $d$-finite dimensional distribution of $X$ has its characteristic function given by $\varphi(s)=e^{\phi(s)}$ whith: $$\phi(s) = ia's - \frac{s'\sigma s}{2}+\int_{\mathbb{R}^d}\big(e^{is'x} - 1 - i s'x \big)\frac{1}{x'x} d\kappa(x), \quad s \in \mathbb{R}^d$$
where $\kappa$ is a measure such that $\kappa(\mathbb{R}^d)< \infty$.
Can I say that $X$ is a Levy process?
Note that it is not obvious that $\varphi_t(s)=e^{t \phi(s)}$. Also, I don't know if this is enough to show that $X$ is a Levy process.
Help!