This is quite quick to prove. Let T be the time of the n’th jump of X. Also let S be a predictable stopping time. As M is a martingale, E[ΔM_{S}]=0 (predictable times are fair). Then, as the jumps of X and M are all of size 1,

So T satisfies the definition of a totally inaccessible stopping time.

Alternatively, use the result that a cadlag increasing integrable process is quasi-left-continuous if and only if its compensator is continuous (quasi-left-continuous = jump times are totally inaccessible). Any textbook which covers similar material to these notes should also include these results.

(apologies for the very slow response here, I’ve been busy lately and not had much time to update the blog).

]]>Thank you for the great explanation. I wondered if you could let me know a textbook or paper where I coulf find a proof for what you say just before Lemma 5, namely: “A consequence of the compensated process {M_t=X_t-\lambda t} being a martingale is that the jump times of X are totally inaccessible”.

Thanks for your help in advance.

All the best.

]]>Ok thanks for this answer

Best regards

Hi. The case where is a continuous deterministic increasing process is relatively simple to deal with, and is the only case where you are assured that the counting process has the Poisson distribution.

The more general case where is just assumed to be continuous, adapted and increasing involves more advanced ideas. Actually, in this case will be the *compensator* of *X*, which is something I mentioned in my recent post on special semimartingales. I am planning on doing a post on compensators (the next post, I think) and I could include a generalization of Theorem 7. The case where is continuous corresponds to *X* being quasi-left-continuous. More generally, you can take to be a right-continuous and increasing predictable process with the constraint ΔΛ ≤ 1 (assuming that *X* can only have jumps of size 1).

I was wondering why you stopped your generalization to inhomogeneous Poisson processes i.e. those for which is a deterministic increasing function, and did not include the case of increasing processes ?

If I put it in another way, what equivalences still hold in theorem 7, if we ask for to be only an increasing process adapted (with respect to some large enough filtration), and maybe how general are those processes in the area of counting processes ?

Best regards

]]>This is one of the best explanation i had come across so far.Self explanatory.Like to receive this article.

Look forward to see in my mail box.

Keep doing best work like this.

All the best.

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