I've got a two-dimensional Markov stochastic process $(X_t, Y_t)$ that runs on time interval $[0, t_f]$. I know the transition function (or the infintesimal generator, if you like) of this process. I'd like to condition it such that $X_t \ge 0$ on $[0, t_f]$ and reform my transition function accordingly.
Is there a general procedure by which this can be done? Or is more information on the behaviour of $(X_t, Y_t)$ necessary?