This is a conversation with Chat-gpt Plus 5.6 Sol. The motivating question from the starting point is a conjecture of Brémaud on whether there exists an anticipating point process which is absolutely continuous with respect to the standard Poisson process. Brémaud's conjecture is No. I tried to intervene when the conversation was losing meaning only twice before it found the right solution. This is the chat history
https://chatgpt.com/share/6a70dd63-2814-83ee-adf8-680d84474b97
Q: What is the current status of Brémaud's 1972 thesis conjecture (p.48) in the field of Poisson processes?
A: I will first pin down exactly which conjecture appears on page 48 of Brémaud's 1972 thesis, and then trace whether it has since been proved, disproved by a counterexample, or only resolved under additional conditions. This is a rather obscure problem, so I will prioritise checking the original thesis/monograph and subsequent citation chains, rather than relying on secondary claims about whether it is still open.
I have found the key point: Brémaud's original conjecture is that there exists no absolutely continuous (with respect to the standard Poisson) anticipative self-exciting process. Under the martingale-intensity definition he gives in the preceding two pages, this original version is actually hit directly by a very classical object - the Poisson bridge / initial enlargement. What really needs to be distinguished is that a later 2025 paper claiming to answer the conjecture adopts a stronger filtration assumption.
Q: But keeping the bounded intensity assumption would distort the original intent.
A: Yes, I agree. And in my previous answer, isolating the "bounded intensity" version as a possible surviving conjecture tends to distort the historical account.
Q: Continue.
A: I will unify all three versions on the canonical point-process space, so that we can directly see: which object is fixed, which filtration is changed, and which notion of "Poisson" is implicitly strengthened. The key issue is not boundedness, but rather that "the Poisson law" and "being a Poisson process with respect to a given filtration" are two distinct propositions and must not be conflated.
Q: But how can a standard Poisson process not depend on a filtration?
A: It can. Here we must strictly separate two levels; your question precisely points out that my previous statement "the standard Poisson process does not depend on a filtration" needs to be made more precise.
Q: OK.
A: I will check the original text on this point: first look at Brémaud's formal definition of "standard Poisson process" in Chapter I, and then see how he invokes P0 and the filtration on p.48. Once these two places match up, the earlier controversy will essentially be settled.
Q: If the same process with respect to the same filtration is not well-defined under two probability measures, does the conjecture even make sense?
A: It does make sense, but "well-defined" must be made concrete. The most common confusion here is between three different things; the first two are what Brémaud's problem really requires, while the third is not needed.
Q: The goal is precisely to identify the differences in characterisation under different probability measures, not to play word games or argue about who is right or wrong.
A: Exactly. What really deserves study is how the Doob-Meyer decomposition of the same (N, G) changes under P0 versus P ≪ P0, and when that change can still be realised by an absolutely continuous density process. Then Brémaud's problem becomes a very standard and meaningful "change of measure + enlargement of filtration" question.
Q: Let us take P0 as the standard Poisson process.
A: Yes. Then we completely fix the framework and stop arguing about whether P0 remains Poisson under the enlarged filtration - that is purely a terminological issue.
Q: Let us not get stuck on standard definitions. Find the correct definition and state the theorem.
A: I agree: we should stop circling around how a "standard Poisson process" should be named. The most natural object is actually a fixed filtered measurable system (Ω, F, G, N), and we compare compensators under different probability measures; "Poisson" is then used only to specify the law of the reference measure P0.
Q: Can we drop the discussion of the Lebesgue decomposition?
A: Yes, and I think dropping the Lebesgue decomposition is actually better. The truly essential notion is the absolute continuity of the predictable compensator as a random measure, without first splitting it into a dt-part, a singular part, and an atomic part.
Q: Is this the correct direction?
A: Yes, this direction is correct, and it is more fundamental than arguing about whether "Poisson process" is defined with respect to a given filtration. But I will tighten it further: the core object is not the Lebesgue decomposition, nor the intensity itself, but the predictable measure class of the compensator.
Q: If the standard process under P0 also requires a Lebesgue decomposition, does that distort Brémaud's original intent in proposing the conjecture?
A: Yes. If we take the compensator of the enlarged filtration under P0 and then perform a Lebesgue decomposition, and treat that decomposition as the main problem, I believe that does begin to deviate from Brémaud's original 1972 conjecture. It can serve as a tool for later obstruction analysis, but it should not be taken as the definitional starting point of the problem.
Q: Focus on Brémaud's specific example and his motivation for the conjecture.
A: I will shift the emphasis back to Brémaud's concrete example on p.48: first reconstruct why he chose that "look ahead by a window of length a" rate, then explain how this example exposes the failure of the classical likelihood construction, and why he would naturally conjecture from this failure that there is no absolutely continuous anticipative self-exciting process.
Q: What is the point of comparing with the standard Poisson process?
A: There are three purposes, and the third is precisely the content of Brémaud's p.48 conjecture.
Q: The path is no longer to be changed; that is a given fact.
A: Yes, that is how we should say it - we must no longer use wording like "change/create paths". The canonical path space and the coordinate process N(ω) never change at all; the only thing that changes is the probability measure on the same set of paths.
Q: How far has the current literature advanced on this step?
A: I will search according to your now-fixed formulation, not by the label "Brémaud conjecture": on the canonical Poisson P0, with the same path space and the same anticipative filtration G, characterise which P ≪ P0 give a prescribed future-dependent λ as a G-intensity. This lets us see whether the existing literature has resolved "change of measure", "enlargement of filtration", or the truly coupled step.
At first glance, the literature has already matured both sides separately: the 1970s Jacod-Kabanov-Liptser-Shiryaev work already gave necessary and sufficient conditions for absolute continuity/singularity of point-process laws in terms of compensators; and post-2000s enlargement-of-filtration literature can compute the compensator after adding future information. What has not been systematically investigated is the combination of these two results specialised to Brémaud's sliding future-window filtration and the classification of realizability of a given anticipative rate.
Q: Conversely, does "there is no anticipating process" mean that every absolutely continuous process is minimal?
A: One can view it in that converse way, and this is actually closer to a "structure theorem". The key is to define "minimal" precisely: not "the intensity takes the smallest numerical value", but rather that, among the filtration families that make N adapted, the predictable intensity cannot be made more essential/smaller by deleting extra information. I will align this equivalence with the definition of minimal predictable intensity used in the text.
There is a crucial logical fork: there exists some anticipative representation and the minimal representation of the process itself is anticipative are not the same thing. The standard Poisson process itself can simultaneously possess a natural minimal representation and an anticipative intensity after enlargement, so if one does not first distinguish these two levels, the converse equivalence will fail.
Indeed, as was pointed out to me by GPT, this is a niche problem with few audience. It attempted to argue with me that the definition in Brémaud's 1972 thesis might be incorrect. It can be encouraging to understand every problem in one's own way, but it gets lost in the abundance of literature. Even worse, it insists on its own formulation and tries to write down solutions despite moving further away from the starting point. I intervened and suggested the correct definition in a questioning tone. Finally, it returned to my recent paper.
I guess this is because the harness in companies requires every artificial intelligence to satisfy users' requests in a very short time. All kindergarten children learn from their teachers the fact that that patience is good. Otherwise you will write poorly, break a cup, and AIs are urged to find the correct answers. For people who are solving open problems, this might take ten years or longer to find such a solution, including: understanding, rephrasing, hypothesising, verifying, analysing, and also rethinking. The abruptness of big data and high-speed computers might change the rules of the last three, but not the first. Patience plays an important role in understanding very difficult problems. Even some problems might need intuition beyond mere logic and conceptual breakthroughs. From this conversation, three main findings are summarised as follows: it will not revoke the things it assumes are already correct; its hypotheses are too strong; it hurries to arrive at a conclusion about true or false. The first two might be remedied, as I believe, in the near future. 今时不论明日事，明日自有明日计。Both of us will take time.