Could you please give me a reference on this stuff?

I need a reference book having results like Lemma 9. Any help will be greatly appreciated.

TheBridge: Thanks for the reference for σ-integrability.

]]>Hi George,

Thank’s for this very neat explanation and for the additional elements.

So here your which tends to almost surely by hypothesis.

The worst part here is that I knew this generalisation of conditional expectation. By the way it is named -Integrability with respect to in He, Wang, and Yan ‘s book “Semimartingale Theory and Stochastic Calculus” (first Chapter).

Best regards

]]>Rather than taking the limit directly, use the fact that to write,

If the indicator function in the front of this expression is moved inside the conditional expectation, then the identity is just using .

Now, you can take the limit as n goes to infinity, and it doesn’t have to be commuted with the expectation at all. You have . Also, as Y is taken to be 0 at infinity, this also holds if the indicator function is changed to . So, .

There is something that I glossed over here, and I was intending to add to the end of this post as a note (I'll do this). I take the conditional expectation without proving that is integrable. In fact, it need not be integrable. For a random variable Z and sub-sigma-algebra , the conditional expectation is always defined, although it could be infinite. If this is almost surely finite, the is well-defined, by for any such that is integrable. Also, is almost-surely finite if and only if there is a sequence with and are integrable. From this, you can show that almost surely, for any locally integrable process *X*. So, makes sense. I’ll append a note to this post along those lines.

Also, I do plan on tidying up these posts and trying to incorporate comments or clarifications but, for now, at least I have your comments at the bottom of the post in case it causes anyone else any confusion. I think the jump in Lemma 2 here is too large for many people to see how this works, so I’m not surprised you picked up on it. Like I said, the original thinking was to add a note concerning conditional expectations at the end and link to it (but, I forgot).

]]>I have another elementary question (I’m afraid) to ask about the proof of point 2 on lemma 2.

There you prove that a.s. for an Integrable Martigale . Right ? *[GL: Correct]*

Applying localization argument here then means that for a localizing sequence of the now only locally integrable, local martingale we have :

a.s. for all right ? *[GL: Correct]*

But letting and noting , we can only conclude that :

a.s.

This is not the statement to be proven and I miss the step that would gives the final conclusion.

Suuch an argument would be :

a.s.

But I can’t justify properly the intertwining of Lim and Expectation operator that gives the conclusion wihtout adding extra assumptions.

Maybe (or probably should I say) I missed something that would trivially lead to the conclusion, so would you please point me that out ?

Best regards

]]>Great !

Now there’s nothing left unclear to me, sorry to be so long to get such elementary details.

Thx for the latex advice and corrections.

Best regards

]]>“for almost every ω ∈ Ω there exists an n such that …” is maybe a clearer way of saying it. That is, n depends on ω. This is implied by almost surely tending to infinity (it’s equivalent to being almost surely unbounded). It can be unclear sometimes exactly what is held fixed and what is dependent on ω, especially when ω is implicit as is usually the case.

Btw, I edited some latex in your post, hope it’s correct now. You have to be especially careful with < and > signs, which can get interpreted as marking HTML tags. The only foolproof way I know is to use the HTML codes < and >.

]]>Hi George,

I am not sure to get exactly what you mean when you say :

“for any finite t>0, we (almost surely) have for large enough n,”

What about for an absolutely continuous random variable of full support over (for example an exponential rv of parameter 1) ?

This sequence is a proper localizing sequence of stopping times as it is increasing almost surely to , but for a fixed time t>0, and any n, we don’t have almost surely that as for any n we have .

I think that it’s this point that has disturbed from the begining and which is why I had to use the contrapositive proposition to convince myself.

Best regards

]]>Yes, that works, although I don’t really think that going to the contrapositive is easier. Just note that, for any finite t, we (almost surely) have for large enough n, and finite variation on , so finite variation on [0,t].

]]>Hi George,

I think I got it this time. To be complete, I ‘ll try to prove the equivalence. The contrapositve proposition is the easiest way to see this for me. So i try to show that X not FV implies X not of Locally Finite Variation.

If is not FV, then there exists an event A of strictly postive probability over which is of infinite variation over some interval [0,t]. In that case, cannot be of Locally Finite Variation, because for any sequence of stopping time increasing almost surely to the stopped process as is of infinite variation over [0,t] conditionally on the set A of strictly positive probability.

Best regards

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