I wonder if the solution to the SDE $$ dX_t= \mu(t,X_t)dt+\sigma(t,X_t)dW_t, $$ for any $\mu, \sigma$ such that this SDE actually has a solution, is progressively measurable.
My line of thought was, that for continuous $\mu,\sigma$ I think it works, since the process then is continuous and adapted (I hope there's no mistake in that thought), but what about more general cases of $\mu$ and $\sigma$? Do I miss a point somewhere to state continuity of $X_t$ in any case?