Definition of intensity of a counting process

Question

Fix a filtered probability space $(\Omega,\mathcal{F},P,(\mathcal{F}_t)_{t\geq 0})$. Let $N=(N_t)_{t\geq 0}$ be a counting process. Then we have the following definitions for the intensity of a counting process: Brémaud gives the following definition in his book "Point processes and queues".

Definition 1: A progressive process $\lambda=(\lambda_t)_{t\geq 0}$ is called the intensity of a counting process if $$\int_0^t\lambda_s ds<\infty$$ for all $t\geq 0$ and $$E\left[\int_0^\infty C_sdN_s\right]=E\left[\int_0^\infty C_s \lambda_s ds\right]$$ for all non-negative $(\mathcal{F}_t)_{t\geq 0}$ predictable processes $C=(C_t)_{t\geq 0}$.

Then on wikipedia, we have the following definition:

Definition 2: Let $N=(N_t)_{t\geq 0}$ be a counting process. Then $N$ is a submartingale and $$M=N-A$$ is a martingale where $A$ is a predictable increasing process. A is called the cummulative intensity of $N$ and if it is of the form $$A_t=\int_0^t\lambda_s ds$$, then $\lambda=(\lambda_t)_{t\geq 0}$ is the intensity of $N$.

On wikipedia, we have also another definition:

Definition 3: The intensity process $\lambda=(\lambda_t)_{t\geq 0}$ is defined by $$\lambda_t=\lim_{h\downarrow 0}\frac{1}{h}E[N_{t+h}-N_t\mid \mathcal{F}_t]$$

Now my question is if these definitions are all equivalent.

For the last definition, we have $$\frac{1}{h}E[N_{t+h}-N_t\mid \mathcal{F}_t]=\frac{1}{h}E[M_{t+h}+A_{t+h}-M_t-A_t\mid \mathcal{F}_t]=\frac{1}{h}E[A_{t+h}-A_t\mid\mathcal{F}_t].$$ Is it true to say that $$\lim_{h\downarrow 0}\frac{1}{h}E[A_{t+h}-A_t\mid\mathcal{F}_t]=E[\lim_{h\downarrow 0}\frac{1}{h}(A_{t+h}-A_t)\mid\mathcal{F}_t].$$ When we have $A_t=\int_0^t\lambda_r ds$, then $$E[\lim_{h\downarrow 0}\frac{1}{h}(A_{t+h}-A_t)\mid\mathcal{F}_t]=E[\lambda_t\mid \mathcal{F}_t]=\lambda_t,$$ which would yield $(2)\implies (3)$.

$(3)\implies(1)$ follows probably by $$E[\int_0^\infty\lambda_sC_s ds]=E[\int_0^\infty\lim_{h\downarrow 0}E[N_{s+h}-N_s\mid\mathcal{F}_s]C_s ds]=E[\int_0^\infty\lim_{h\downarrow 0}(N_{s+h}-N_s) C_sds]=E[\int_0^\infty N_s'C_sds]=E[\int_0^\infty C_s dN_s]$$

Answer 1

I've been trying to find out something similar. As far as I can tell, this is not necessarily true. For example, let $X$ be a Markov process on the state space $\{0,1,2\}$ with intensity matrix $Q$. Assume all elements of $Q$ are finite. Let $Y$ be the process that counts transitions of $X$ to state 2 with respect to the filtration $\mathcal{F}_t = \sigma(X[0,t])$.

By definition 3, it's simple to show that $\lambda_t = \mathbb{I}_{X_t\neq 2}q_{X_t,2}$, which is not generally predictable.

By definition 2, if $\lambda_t$ is an intensity of $Y$, then so is $\tilde{\lambda}_t := \lim_{s\nearrow t} \lambda_s$. So $\tilde{\lambda}_t$ is a predictable process that is an intensity of $Y$ by definition 2. But we've already shown that the intensity defined by definition 3 is not predictable.