Is Theorem 9 remain valid for time-inhomgeneous Markov proccess with a feller evolution systems? ]]>

A question: from Monroe’s Theorem we can represent a cadlag semimartingale as a time-changed Brownian motion (omitting some of the statement). If I understand the theorem, a Poisson process should be representable as a time-changed Brownian motion (as should a compound Poisson process). Has the explicit time change appeared in print? ]]>

https://math.stackexchange.com/questions/4262829/path-of-a-particle-with-brownian-heading

]]>First of all, you have an amazing blog, thank you very much for your work!

Apart from that, I have some trouble with the proof of Theorem 8.

Basically, we take a monotone class generated by $\mathcal E$ and prove that it is closed under intersections and complements. However, as far as I understand, one feature is still missing: it is never mentioned that the set $E$ belongs to the corresponding monotone class. In Theorem 7 it holds automatically (due to $\mathcal E$ being an algebra), but how can we say that $E$ is in $M(\mathcal E)$?

Thank you in advance!

]]>thanks a lot for this amazing work! I do have a question through regarding your statement that left-continuous adapted processes which are almost surely bounded on every finite time interval are automatically locally bounded. I have no problem with that statement whenever the underlying filtration is right-continuous, as I can use first-hitting times of say [n,\infty) to localise, but those would typically fail to be stopping times when the filtration is not right-continuous.

I tried to think of a counter-example: take a process which is 0 until time 1 (included) and then equal to some unbounded random variable Y after (strictly) time 1. Take F to be the natural filtration of that process, which is not right-continuous unless I am mistaken. The process is càglàd, and I expect that one can describe more or less explicitly any F-stopping time as being either deterministic or some measurable function f(Y), with f valued in (1,\infty), and I haven’t been able to create any localising sequence from this. But I could of course be missing something!

I am mainly asking as you’re using in a later post the fact that left-limits of F-adapted càdlàg processes are always locally bounded and thus in L^1(X) for any F-semimartingale X, and it could be that this statement either requires right-continuous filtrations, or maybe a different argument.

Would be happy to hear your thoughts on that!

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