Now, doing things right thanks to some comments (sorry for upsetting some people) I would like to share my doubt.Attempting to calibrate Heston plus jumps model through markovian projection using gatheral ansatz referring some conditional expectations, I faced some problems. At this point I would like to express it in an analog and simpler way, because perhaps it would give me the clearness to attempt and solving my original problem.It simply is: What is the meaning and how could I compute [$]E[ dW_t | S_T][$] where [$] t<T[$] and [$]\frac{dS_t}{S_t}= \mu_t dt + \sigma_t dW_t [$]Thank you very much and excuse the past trouble.

Why would you want to use markovian projection for Bates (heston + jumps)? I mean, isn't there a characteristic function for this model? Unless you are using some funky jump process of course..