Poisson Process Change of Measure I have seen the following result stated in the literature:
Let $N(t)$ be a (finite time horizon) Poisson process defined on a probability space $(\Omega, \mathbb{P})$ with constant intensity $\lambda$. There exists an equivalent probability measure $\mathbb{Q}$ such that $N(t)$ has intensity $\mu \neq \lambda$, i.e. we can change constant intensity by a change of measure.
My question is whether or not it is possible to change the intensity from being constant to being random (i.e. stochastic) under an equivalent change of measure?
I have seen that there is a Girsanov theorem for jump processes which seems to suggest that this is possible. Unfortunately, I do not have a deep enough understanding of the stochastic analysis to know if all the technical conditions hold. I would also be greatful if anyone could point me to accessible literature on this topic. 
Many thanks.
 A: Yes. For example, restricted to $[0,T]$, the density the process with random intensity $Z$ (also known as a Cox process with directed by $Z$) with respect to a intensity one Poisson process can be constructed as follows: Let $\{t_i\}_{i \in N}$ be an increasing sequence of positive reals, and z a positive real valued function. We define the functional
$$G(z, \{t_i\}_{i \in N}) = \exp\left(T - \int_0^T z(s) ds + \sum_{0 \leq t_i \leq T} \log(z(t_i))\right) $$
Let $\{J_i\}_{i \in N}$ be the sequence of jump times of a Poisson process then given a realization of a unitary Poisson process then the density with that converts this process in a Cox process directed by $Z$ is 
$$ E(G(Z,\{J_i\}_{i \in N})\, | \,\{J_i\}_{i \in N}) = E(G(Z,\{t_i\}_{i \in N})) \bigg|_{\{t_i\}_{i \in N} = \{ J_i \}_{i \in N}} $$
For example the density of a Poisson process of constant intensity $\lambda$ with respect a unit rate Poisson process $N$ is:
$$\exp\left( T - \lambda T + \log(\lambda) N_T\right)$$
You can check that integrating with this density the characteristic function you will arrive to the Poisson process with intensity $\lambda$
