Looking for Cadlag Modification I stumbled over this blog post on almostsure.wordpress.com.

I would Like to use the following theorem form that post:

Theorem: Let X be a martingale, submartingale or supermartingale which is right-continuous in probability. Then, it has a cadlag version.

Can someone hint me to a reference in the mathematics Literature, where I can find this result?

  • $\begingroup$ @TheBridge nice joke! However, I'd like to know a peer-reviewed reference. $\endgroup$
    – Vincent.W.
    Apr 19, 2018 at 12:05
  • $\begingroup$ May be you could try copy/paste the marvelously clear proof of George Lowther on his blog (which is self contained what else to ask for ?), otherwise theorem 9 by P. Protter page 8 of its book on stochastic integration is an alternative but Protter is generally more expeditious in its proofs that sometimes lack details in my opinion. Karatzas and Shreve's book on BM also has a proof see theorem 3.13 page 16. Once again Lowther's proof is the clearest and this is no joke. $\endgroup$
    – The Bridge
    Apr 19, 2018 at 13:34
  • $\begingroup$ @TheBridge Yeah I agree, and I use his Blog a lot. But I will not cite a personel Blog. The references you named don't quite fit the Theorem, as they assume right-continity of the filtration. $\endgroup$
    – Vincent.W.
    Apr 19, 2018 at 14:32
  • $\begingroup$ I am realizing that the theorem that you mention do not establish that the modification fulfills the martingale property. I have posted a comment on George Lowther's blog about this . The most general result in the classical literature that I could found is theorem 4.2 - people.math.ethz.ch/~jteichma/martingales_finance_20150610.pdf ( but it's not what you are looking for). In a another form known as Fölmer's lemma is theorem 2.44 (He Wang Yang - Semeimartingale theory and Stochastic Calculus). In the end this post was richer than I thought. Thank you. $\endgroup$
    – The Bridge
    Apr 19, 2018 at 16:06
  • $\begingroup$ The Follmer's lemma is here : math.stackexchange.com/questions/1020050/… $\endgroup$
    – The Bridge
    Apr 19, 2018 at 16:11


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