Let $X_t$ be the compound Poisson process
$$
X_t = t - \sum_{i=1}^{N_t} \xi_i, \tag{1}
$$
where $N$ is a Poisson process with rate $\lambda$ and the $\xi_i$ are i.i.d., positive, with common distribution $F$.
The characteristic exponent of $X_t$ is
$$
\Psi(\theta) = \theta - \lambda \int_{(0, \infty)} \left( 1 - e^{-\theta x}\right)F(dx). \tag{2}
$$
Assume $\Psi$ has a root $\theta^* \ne 0$. Define the stopping time
$$
\tau = \inf_{t > 0} \left\{ X_t > x \right\} , x > 0. \tag{3}
$$
Show
$$
E\left[\exp\left(\theta^*X_\tau -\Psi(\theta^*)\tau\right)1_{\{\tau < \infty\}}\right] = e^{\theta^*x}P\left( \tau < \infty \right). \tag{4}
$$
This is exercise 1.9 in Kyprianou, Fluctuation of Levy Process.
The solution is given, but I don't see why (4) holds.
As $\theta^*$ is a root, the left of (4) reduce to
$$
E\left[\exp\left(\theta^*X_\tau \right)1_{\{\tau < \infty\}}\right].
$$
But, by (2), $X_t$ is a jump process. So, I cannot infer that
$$
X_\tau = x, \text{a.s.}
$$
