I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory.
I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and Shreve, for example).
My PDE theory is pretty weak. I know about the Fokker-Planck equations, and that's about it. My guess is that before I begin reading, I'll need to learn a significant amount of classical PDE theory. I would appreciate any recommendations for PDE textbooks that emphasize material that will be useful in the study of SPDEs. If someone could recommend a gentle introduction to SPDEs to go with it, I would be very grateful.