Lévy processes and subordinators

Prove that the characteristic function $\phi (t, \theta ) = \mathbb {E}\left[e^{i \theta X(t)}\right]$ of a Lévy process $X$ is a continuous function of $t$.

Show that a compound Poisson process is a Lévy process, and find its Lévy symbol.

Verify the formula of Example (8.6.6) for the Laplace exponent of the Moran gamma process.

Prove Theorem (8.6.7): the property of being a Lévy process is preserved under time change by an independent subordinator.

Let $N$ be a Poisson process of rate $1$, and $T$ an independent random variable with the $\Gamma (1, t)$ distribution. Show that $Y = N(T)$ has probability generating function $\mathbb {E}\left[s^{Y}\right] = (2-s)^{-t}$ for $s < 2$.

Let $X$ be a Lévy process with finite variances. Show that the following are martingales:

Let $X$ be a continuous-time martingale and $T$ an independent subordinator. Show that, subject to the moment condition $\mathbb {E}\left[\left|Y(t)\right|\right] < \infty , Y(t) = X(T(t))$ defines a martingale with respect to a suitable filtration. Show that the moment condition is valid whenever $X$ is a positive martingale.