MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$(X_t, Y_t)$ 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?

share|cite|improve this question
Have you tried applying Doob's h-transforms? – Bati Jan 8 '13 at 18:10

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 $(Y_t)_t$.

share|cite|improve this answer

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.