Deterministic control of SDEs with stochastic drift and multiplicative noise: a variational approach

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.