Let $(\mathcal F_t)$ be a filtration and $X$ a random variable such that $E[X | \mathcal F_t] \neq X$ and $E[X | \mathcal F_t] \neq E[X]$ for all $t\in [a,b]$. We consider the conditional expectation
$$E[X | \mathcal F_t] $$
How do we simulate paths of the above process when there is no $Y_t$ such that $E[X | \mathcal F_t]=E[X | Y_t]$ ?