  We consider a linear stochastic differential equation with stochastic drift.
We study the problem of approximating the solution of such equation through an
Ornstein-Uhlenbeck type process, by using direct methods of calculus of
variations. We show that general power cost functionals satisfy the conditions
for existence and uniqueness of the approximation. We provide some examples of
general interest and we give bounds on the goodness of the corresponding
approximations. Finally, we focus on a model of a neuron embedded in a simple
network and we study the approximation of its activity, by exploiting the
aforementioned results.
