The integral of the squared Gaussian process

This work studies the random variable defined by X≔∫_t^TZ_s^′AZ_sds, with A a real matrix of size N×N, and Z_s∈R^N Gaussian processes. The results show that X is a constant variable when Z_s is time-independent. When Z_s∈R follows a Brownian motion, a closed-form moment generating function (MGF) of X is derived, which does not match the MGFs of known distributions. Finally, a portfolio problem is presented to show how the MGF of X is needed for finding the optimal solution in closed form.