Conditional probability of a general Markov process given by its running process I have a question as follow:
"Let $X$ be a general Markov process, $M$ is a running maximum process of $X$ and $T$ be an exponential distribution, independent of $X$. 
I learned that there is the following result:
Probability: $P_x(X_T\in dz \mid M_T=y)$ is independent of starting point $x$ of the process $X$. Where $y, z \in R$"
Is there anyone who knows some references which mentioned the result above? I heard that this result was found around the seventies but I haven't found any good reference yet. 
Thanks a lot!

For real-valued diffusion processes, this is essentially a local form of David Williams' path decomposition, and can be deduced from 
Theorem A in a paper "On the joint distribution of the maximum and its location for a linear diffusion"  by Csaki, Foldes and Salminen 
[Ann. Inst. H. Poincare Probab. Statist., vol. 23 (1987) pp. 179--194].
For more general Markov processes, you will need to look into the theory of "last-exit times".  Although these are not stopping times, many Markov processes possess a sort of strong Markov property at such times. This theory can be applied to the last time before $T$ that
the process is at level $y$.  One place to start might be the paper of Meyer, Smythe and Walsh "Birth and death of Markov processes" in vol. III (pp. 295-305) of the Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (1972).
See also the work of P.W. Millar from roughly the same time period.
