# Is there a standard procedure for conditioning a stochastic process?

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?

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Consider a homogenous Markov process $(X_t)_{t\geqslant0}$ in continuous time with state space $\mathbb X$ and infinitesimal generator $A$. Consider the first hitting time $\tau_0=\inf\{t\geqslant0\mid X_t\in\mathbb X_0\}$ of some subset $\mathbb X_0$ of $\mathbb X$. For every $t\geqslant0$ and $x$ in $\mathbb X$, define $$h_t(x)=\mathbb P_x(\tau_0\geqslant t).$$ For every $s\gt0$, the process $(X_t)_{0\leqslant t\lt s}$ conditioned on $\tau_0\geqslant s$ is an inhomogenous Markov process whose infinitesimal generator $A^s_t$ at any time $0\leqslant t\lt s$ is defined by $$A^s_t(u)=(h_{s-t})^{-1}\cdot A(h_{s-t}\cdot u).$$