= Solution
Use the intrinsic definition of a <Lévy process>: $X_0=0$ <almost surely>, its increments over disjoint time intervals are independent, their <probability distributions> depend only on interval length, and $X$ has <stochastic continuity>. In symbols,
$$
X_{s+t}-X_s\overset d=X_t,\qquad X_{t+h}\longrightarrow X_t\text{ in probability as }h\to0
$$
with times restricted to the half-line. Such a <stochastic process> has a <càdlàg modification>, and is usually represented by that version. Requiring <càdlàg> paths in the definition is a common equivalent convention at the level of modifications; it is important to distinguish this from a claim about the paths of an arbitrary supplied version.
For the <characteristic function>, write
$$
\boxed{\mathbb E e^{i\theta X_t}=e^{t\psi(\theta)}=e^{-t\Psi(\theta)},\qquad\Psi=-\psi.}
$$
Here $\psi(0)=0$ and $\psi$ is the positive-time-sign <characteristic exponent of a Lévy process>. The <independent increments> and <stationary increments> give $\phi_{s+t}(\theta)=\phi_s(\theta)\phi_t(\theta)$; <stochastic continuity> gives continuity in time and $\phi_0=1$. This continuous multiplicative <semigroup> has the stated exponential form. Its exponent has the <Lévy–Khintchine formula>
$$
\psi(\theta)=ib\theta-\frac{\sigma^2\theta^2}{2}+\int_{\mathbb R\setminus\{0\}}\left(e^{i\theta y}-1-i\theta y\mathbf1_{\{|y|\leq1\}}\right)\nu(dy),
$$
where $b\in\mathbb R$, $\sigma^2\geq0$, and the <Lévy measure> $\nu$ satisfies $\int(1\wedge y^2)\nu(dy)<\infty$. The truncation convention fixes the <drift coefficient> $b$; the displayed sign convention agrees with $\Psi=-\psi$.
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