= Solution
A real <Lévy process> is a <stochastic process> $(X_t)_{t\geq0}$ with $X_0=0$ almost surely, <independent increments>, <stationary increments>, and <stochastic continuity>. The latter means $X_t-X_s\to0$ in probability as $t\to s$. Its usual path convention is <càdlàg>: paths are right-continuous and have left limits. Equivalently one starts with the increment and continuity-in-probability conditions and then chooses the càdlàg version.
The <Lévy characteristic exponent> is specified here by
$$
\boxed{\mathbb E e^{i\theta X_t}=e^{-t\psi(\theta)}.}
$$
Some conventions use $e^{t\widetilde\psi(\theta)}$ instead, with $\widetilde\psi=-\psi$. \b[The sign convention must be fixed before computing the exponent.] The zero process satisfies this definition; excluding it requires an additional nondegeneracy condition.
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