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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!

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What do you call a running process of a Markov process? –  Did Dec 27 '11 at 17:44
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Hello Didier Piau, $M_t:= \sup \limits_{s \leq t}X_s $ is called a running process of $X$. –  ltt Dec 27 '11 at 19:26
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Clarify your central expression please. P(X) should be P(X = x) or P(X <=x) or something similar (the first term in parenthesis), depending on whether it's discrete or continuous. Next, this certainly doesn't hold in all generality. Imagine a constant process. Any such process is Markovian, and will have X and M constant as well, but at different values, so their probability measures will be very different. –  gnometorule Dec 27 '11 at 19:58
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@gnometorule: I have edited the post. I hope it is clearer. –  ltt Dec 28 '11 at 0:06

1 Answer 1

up vote 3 down vote accepted

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.

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Thanks a lot @John Dawkins. I have also found the paper from Millar: projecteuclid.org/… –  ltt Dec 28 '11 at 10:41
    
+1. Nice bibliographic search. I wonder if, once the research papers you indicate were digested by the community, this result got explained in some textbooks as well... –  Did Dec 28 '11 at 15:44

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