  Distributional properties -including Laplace transforms- of integrals of
Markov processes received a lot of attention in the literature. In this paper,
we complete existing results in several ways. First, we provide the analytical
solution to the most general form of Gaussian processes (with non-stationary
increments) solving a stochastic differential equation. We further derive the
characteristic function of integrals of L\'evy-processes and L\'evy driven
Ornstein-Uhlenbeck processes with time-inhomogeneous coefficients based on the
characteristic exponent of the corresponding stochastic integral. This yields a
two-dimensional integral which can be solved explicitly in a lot of cases. This
applies to integrals of compound Poisson processes, whose characteristic
function can then be obtained in a much easier way than using joint
conditioning on jump times. Closed form expressions are given for
gamma-distributed jump sizes as an example.
