Deterministic Control of SDEs with Stochastic Drift and Multiplicative Noise: A Variational Approach

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