I finally have the generalization that will show that randomized stopping can’t give better results than optimal stopping. It can be found here. While in the previous post I only claimed I can do it for step functions, here I show that it can be done for any continous and adapted function satisfying a basic integrability condition. I think one could also drop the assumption on continuity. This could probably be done in the same way as it is done in the construction of the Ito integral. I don’t think I’ll have to time to look into that any time soon though.
Generalizing the previous result, step 2