This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.
"Controlled diffusion processes." Probab. Surveys 2 213 - 244, 2005. https://doi.org/10.1214/154957805100000131