Integrated stationary Ornstein-Uhlenbeck process, and double integral processes

We find a representation of the integral of the stationary Ornstein-Uhlenbeck (ISOU) process in terms of Brownian motion Bt; moreover, we show that, under certain conditions on the functions f and g, the double integral process (DIP) D(t)=∫βtg(s)∫αsf(u)dBuds 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.