$(X_t, Y_t)$ is a two-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. Given its transition function $a(x, y | x', y')$, I would like to condition the process on $\inf_{s \in [0, t_f]} X_s \ge k$ and find the new transition function.
Can the problem be solved at this level of generality? Or must we dig into the specifics of $a$ to find a solution on a case-by-case basis?
More generally, let $A$ denote some subset of the state space of a homogenous Markov process $(Z_t)_{0\leqslant t\leqslant t_0}$ with transition kernel $(a_{t-s})_{0\leqslant s\leqslant t\leqslant t_0}$, $T=\inf\{0\leqslant t\leqslant t_0\mid Z_t\in A\}$, $\mathbb Q=\mathbb P(\ \mid T=+\infty)$, and $h_t(z)=\mathbb P(T\gt t\mid Z_0=z)$. By an elementary conditioning, with respect to $\mathbb Q$, the process $(Z_t)_{0\leqslant t\leqslant t_0}$ has inhomogenous transition kernel $$ \mathbb Q(Z_t=z\mid Z_s=z')=a_{t-s}(z\mid z')h_{t_0-t}(z)h_{t_0-s}(z')^{-1}, $$ for every $0\leqslant s\leqslant t\leqslant t_0$ and every $z$ and $z'$ not in $A$.
Apply this to $Z=(X,Y)$ and $A=(-\infty,k)\times\mathfrak Y$, where $\mathfrak Y$ is the state space of $(Y_t)_t$.
www.math.harvard.edu/~alexb/rm/Doob.pdf
