Does point process ordering ever imply conditional intensity ordering?

Question

Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\mathbb N}$. They are regular/non-explosive in the sense that they always have a finite number of points in any finite interval. Assume both processes have predictable conditional intensity functions (CIFs) $\lambda(t)$ and $\lambda'(t)$. See Section 7.2 in Daley & Vere-Jones (2003), An introduction to the theory of point processes. Vol. I for more details about the conditional intensity function (Def. 7.2.II). Note that there is a question of uniqueness of CIF here, but that is taken care of by taking the left-continuous version, by definition (this is discussed in the text referenced above).

Note that the CIF actually involves implicitly conditioning on the history of the point process up to time $t$ (but not including $t$, technically). The CIF is in general a stochastic process itself, depending on the particulars of the point process, e.g. a Poisson process whose intensity randomly jumps between several values (such as a Markov-modulated Poisson process).

I am interested in stochastic domination where $N\subset_{st} N'$, i.e. that we can construct a coupling so that $N'$ has a point wherever $N$ does (almost surely). See Section 2.9 in Szekli (1995) Stochastic ordering and dependence in applied probability for more details on such stochastic domination.

It is well-known that domination of CIFs implies domination of point processes. In other words, if $\lambda(t)\leq\lambda'(t)$ for all $t\geq0$ (note that this is actually a stochastic domination as well) implies that $N\subset_{st} N'$. See Theorem E in Szekli (1995) linked above or Theorem 2 in Rolski & Szekli (1991), Stochastic ordering and thinning of point processes.

My question is: does $N\subset_{st} N'$ ever imply $\lambda(t)\leq\lambda'(t)$ for all $t\geq0$?

I can't find any comments on this converse statement at all. Now I can imagine that there are all sorts of complicated processes for which this converse is false (the answer to my question is negative). I would love to have some explicit examples given. But I feel like there must be examples where the implication goes both ways. The trivial example is Poisson point processes with constant intensities (yes?). It seems that comparing a Poisson point process to one with a non-homogeneous but nonrandom intensity would also give the implication going both ways. What about comparing a Poisson point process (with constant intensity) to one with a fairly well-behaved random intensity that is a continuous function but just has jumps at arrivals (like a self-excited process)? It seems to me like point process domination requires domination of such well-behaved intensities (at least when consider stochastic orderings with basic Poisson point processes).

So my claim is: Let $N$ be a Poisson point process with constant intensity $\alpha$ and let $N'$ be a point process with predictable CIF $\lambda(t)$. Then $N\subset_{st} N'$ if and only if $\alpha\leq\lambda(t)$ for all $t>0$ and all possible histories of arrivals.

My attempt at a proof sketch is that if $\alpha>\lambda(t)$ for some $t$ (and some history of arrivals), then $\alpha>\lambda(s)$ for $s\in (t-\epsilon,t)$ for some $\epsilon>0$. This is because the CIF is left-continuous and there are finitely many points in $[0,t]$. Hence, we cannot get point process domination since $N$ has a higher probability of having at least one point in $(t-\epsilon,t)$.

Am I missing something?

Answer 1

I am placing this as a partial answer here rather than a new question or editing the question.

I have found a counterexample where $N\subset_{st}N'$ but the conditional intensities are not ordered. I modified the example found here: https://mathoverflow.net/a/420728/68851 by converting it to continuous-time and ensuring that there were not atoms of positive probability.

Let $(N,N')$ be coupled point processes on $[0,1]$ so that $N$ is a Poisson process with constant intensity $1$. Let $N'$ simply be identical to $N$ unless $N$ has no points, in which case, generate a single point uniformly over $[0,1]$. Thus, $N'$ will always have at least one point.

It is obvious that $N'$ dominates $N$. It is also intuitively straightforward to see that the conditional intensities do not necessarily stay ordered.

There are three cases of histories. I'll change the notation a bit to stay more aligned with that of Daley & Vere-Jones. Let $\tau_j$ be the $j^{th}$ arrival time (not the interarrival time). Now we have $\mathcal H_{t}=\{\tau_1=t_1,\ldots,\tau_{n-1}=t_{n-1},\tau_{n}\geq t\}$ which is the history of arrivals up to by not including time $t$, requiring that $t_1<t_2<\cdots$, etc. Notice that this event includes the fact that the next arrival $\tau_{n}$ could occur exactly at time $t$, but the CIF will not see that arrival actually be design.

Obviously $\lambda(t|\mathcal H_{t-})=1$ since $N$ is a Poisson process with constant intensity.

For $N'$ though, $\lambda'(t|\mathcal H_{t-})$ will be an weighted average of the uniform intensity and the Poisson intensity though. If $\tau_2<t$ it follows that $\lambda'(t|\mathcal H_{t-})=1$ since we cannot possibly be in the 'uniform regime.' If $\tau_1\geq t$, then we will be averaging between the uniform intensity and the Poisson intensity. We can show that the uniform intensity is always greater than 1, hence, this average intensity will be greater than 1 as well. So if we have no arrivals or two or more arrivals in our history, then there is no problem for realizing the domination vie CIFs.

It is $\mathcal H_{t-}=\{\tau_1=t_1,\tau_2\geq t\}$ that causes the problem. This is because, once the first arrival occurs we could still be in the 'uniform regime' which now has intensity zero since it will definitely not produce anymore points. So we are averaging between the constant Poisson intensity of 1 and the, now zero, uniform conditional intensity. Hence $\lambda'(t|\tau_1=t_1,\tau_2\geq t)<1$ for such a conditional history.

Deriving the precise formulae was quite difficult for me as I kept making errors. I ended up calculating them various ways: (1) discretizing the process and using weak convergence, (2) directly computer conditional survivor functions and using $$ \lambda(t)|\mathcal H_{t-})=\frac{-\frac{d}{dt}S(t|\mathcal H_{t-})}{S(t|\mathcal H_{t-})}, $$ and (3) using $$\lambda'(t|\mathcal H_{t-})=\lim_{h\to0} \frac1h\mathbb E[N'_c(t,t+h)\mid \mathcal H_{t-}]$$ where $N'_c(I)$ is the number of points in interval $I$. The weak convergence should be ok since I can argue we are working with counting processes and the Skorohod topology gives all the desired weak convergence theory needed.

I won't include the computations here unless somebody asks for it though.

The conditional intensity I computed for $N'$ is $$ \lambda'(t|\mathcal H_{t-})= \begin{cases} \frac{1+e^{t-1}}{1-te^{t-1}} &\text{ if } \ \mathcal H_{t}=\{\tau_1\geq t\}\\ \frac{1}{1+e^{t-1}} &\text{ if } \ \mathcal H_{t}=\{\tau_1=t_1,\tau_2\geq t\}\\ 1 &\text{ if } \ \mathcal H_{t}=\{\tau_1=t_1,\tau_2=t_2\}\\ \end{cases}. $$

I verified this was consistent with numerical simulations as well in numerous different ways too (e.g. by fitting Survivor functions and , and it seems to work.

The uniform conditional intensity is not too hard to derive on its own and is $$\lambda^\textrm{unif}(t|\mathcal H_{t-})= \begin{cases} \frac{1}{1-t} &\text{ if } \ \mathcal H_{t}=\{\tau_1\geq t\}\\ 0 &\text{ if } \ \mathcal H_{t}=\{\tau_1=t_1,\tau_2\geq t\}\\ \end{cases}.$$ Note that, when conditioned on no point having yet occurred, this diverges to $+\infty$ as $t\to1$ which is intuitive since a point is required to occur and it must therefor become more intense as the end of time approaches.

Now, as for finding some criteria for when point process domination implies CIF domination, I don't have an answer yet, but I do have some preliminary thoughts at least. My instinct is that if the (random) intensity if a Markov process, then the domination result does indeed go both ways. I'll keep working on it, but it would be great for somebody to chime in with some insight.