Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

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?

share|cite|improve this question
up vote 2 down vote accepted

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 $$ A^s_t(u)=(h_{s-t})^{-1}\cdot A(h_{s-t}\cdot u). $$

share|cite|improve this answer
did, you're awesome, you're basically teaching me stochastic processes in your free time. – GMB Jan 8 '13 at 9:09
By the way this kind of conditioning is most of the time refered as the Doob h-transform you can have an intuitve presentation here : – TheBridge Jan 8 '13 at 12:18

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.