  We consider a linear stochastic differential equation with stochastic drift
and multiplicative noise. We study the problem of approximating its solution
with the process that solves the equation where the possibly stochastic drift
is replaced by a deterministic function. To do this, we use a combination of
deterministic Pontryagin's maximum principle approach and direct methods of
calculus of variations. We find necessary and sufficient conditions for a
function $u \in L^1(0,T)$ to be a minimizer of a certain cost functional. To
overcome the problem of the existence of such minimizer, we also consider
suitable families of penalized coercive cost functionals. Finally, we consider
the important example of the quadratic cost functional, showing that the
expected value of the drift component is not always the best choice in the mean
squared error approximation.
