Thanks for the detailed notes here! I wonder whether there is any work on distinguishing two stochastic processes in the collection of r.v.’s sense. What technique and what condition should we have in order to uniquely determine a stochastic process that is continuous in the r.v. sense?

]]>Great post, I find your blog touch on a wide range of topics most conventional textbooks don’t cover. In fact that’s how I came about this website looking for results working on left right limit processes (more specifically optional semimartingales). I’m wondering if you can point me towards any textbooks or papers which discusses/proves the results in this post on stochastic processes with left right limits and also textbooks that discusses in depth predictive, optional, progressive measurable and adapted processes. Thanks in advance!

Sam

]]>No problem. Also I fixed your latex (https://almostsure.wordpress.com/using-latex-in-comments/)

]]>Aha as I consider how stopping times work, specifically the description in the Stopping Times section, I retract my question. Further apologies for the spam.

]]>Also, apologies, I’ve no idea how to get the TeX markup to work…

]]>First, this blog is much appreciated. I couldn’t believe my luck finding such a comprehensive and easy-to-understand compendium on a topic that seems to be pretty scarcely populated with resources. I’m working my way through to your treatment of Quasi-martingales because the only other resource I’ve found on discrete-time quasi-martingales is by Metivier, and his book is inaccessible.

Second, I have a (perhaps naive) question. How is in lemma (3) a function of ? I can see taking an input and returning a value in , but itself shouldn’t necessarily take any inputs unless I’m missing something entirely?

Thank you again,

David

1) The point of restricting the statement to right-continuous (or left-continuous) processes is that, for X to equal Y everywhere, it is only necessary to show that this holds on the rationals. Taking right-limits extends the equality to irrational t. And, the rationals are countable, so the processes will be equal on the rationals, almost surely.

2) We have two sigma algebras. The Borel sigma algebra on the positive reals , and the sigma algebra on the underlying probability space. A (real) stochastic process is a map . measurable w.r.t. the product sigma algebra means measurable with respect to in the domain and in the codomain.