The conditional distribution of a Ornstein-Uhlenbeck $X(t)$ conditional on $X(0)$ is given by

$$ X(t)|X(0) = X(0)e^{-t} + \mu(1 - e^{-t}) $$

This process is usually only defined for $t>0$ (future values), but I see no reason why this shouldn't extend backwards as well:

$$ X(|t|)|X(0) = X(0)e^{-|t|} + \mu(1 - e^{-|t|}) $$

However, I was wondering if this can also be extended between two points as well. I know this is done in kriging, but I was wondering if something similar can be applicable here as well:

$$ X(t)|X(a),X(b) = \left\{\begin{matrix} X(a)e^{-(a-t)} + \mu(1 - e^{-(a-t)}) & t < a \\ \color{red}{???} & a < t < b \\ X(b)e^{-(t-b)} + \mu(1 - e^{-(t-b)}) & b < t \end{matrix}\right. $$


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