If we have an inhomogeneous Poisson process with intensity $\lambda(t)$, what does the covariance function $\mathbb{E}[X_s, X_t]$ look like? Can anyone point me to a derivation?
I would like to ask the same question for a Hawkes type process, where the intensity can be "level-dependent".
If these question have a complicated answer, what I am really interested in is that I have an entire covariance matrix $\Sigma(t, s)$ for a set of discrete $t$ and $s$ and I would like to use it to estimate $\lambda(t)$.
Thank you!
For an inhomogeneous Poisson process with intensity $\lambda(t)$, $X(t)$ (representing the number of occurrences in the interval $[0, t]$) is a Poisson random variable with parameter $\Lambda(t) = \int_0^x \lambda(x)\; dx$. The covariance is $\text{Cov}(X_s, X_t)$. But the numbers of occurrences in disjoint intervals are independent. Thus if $s \le t$, $X(s)$ and $X(t) - X(s)$ are independent, and
$$\text{Cov}(X(s), X(t)) = \text{Cov}(X(s), X(s)) + \text{Cov}(X(s),X(t) - X(s)) = \text{Cov}(X(s),X(s)) = \text{Var}(X(s)) = \Lambda(s)$$