Abstract We find a representation of the integral of the stationary Ornstein–Uhlenbeck (ISOU) process in terms of Brownian motion B t ; moreover, we show that, under certain conditions on the functions f and g , the double integral process (DIP) D ( t ) = ∫ β t g ( s ) ∫ α s f ( u ) d B u d s can be thought as the integral of a suitable Gauss–Markov process. Some theoretical and application details are given, among them we provide a simulation formula based on that representation by which sample paths, probability densities and first passage times of the ISOU process are obtained; the first-passage times of the DIP are also studied.