Characteristic function of time-inhomogeneous Lévy-driven Ornstein–Uhlenbeck processes

We derive the characteristic function (CF) of integrals of Lévy-driven Ornstein–Uhlenbeck processes with time-inhomogeneous coefficients. The resulting expression takes the form of the exponential integral of the time-changed characteristic exponent. This result is applied to some examples leading to closed form expressions. In particular, it drastically simplifies the calculations of the CF of integrated Compound Poisson processes compared to the standard approach relying on joint conditioning on inter-arrival jump times.