Let $(\Omega,\mathcal{F},P)$ be a probability space. Moreover let $\tau_x\colon \Omega \to \Omega$ for $x \in \mathbb{Z}$ be an ergodic group of tranformations that preserves $P$. By ergodic we mean that, up to sets of measure $0$, the only invariant measurable sets under the action of $(\tau_x)_{x \in \mathbb{Z}}$ are $\emptyset$ and $\Omega$.
Let $p_1,p_{-1}\colon \Omega \to [0,+\infty)$, $x \in \mathbb{Z}$ be two bounded and nonnegative functions such that $$ p_1(\omega) + p_{-1}(\omega) = p_{-1}(\tau_1 \omega) + p_1(\tau_{-1} \omega) $$ for almost every $\omega$.
Assume that $X(t)$ is a continuous time Markov process with state space $\mathbb{Z}$.
We define now a random walk $X_t^\omega$, $t \geq 0$ (on some other probability space) with jump rate from $y$ to $y+1$ (resp. from $y$ to $y-1$) $$ p(y,y+1,\omega) := p_1(\omega) \quad \text{resp}.\ p(y,y-1,\omega) := p_{-1}(\omega) $$ for $y \in \mathbb{Z}$.
Finally, we introduce environment process $$ \eta(t) := \tau_{X_t^\omega}(\omega). $$ It is not so hard to prove that $\eta$ is a Markov process.
Now we can write $$ X_t^\omega = N_1(t) - N_{-1}(t), $$ where $N_1(t)$ (resp. $N_{-1}(t))$ is the number of times that the process $\eta(t)$ jumped from a state $\eta$ to $\tau_1 \eta$ (resp. $\tau_{-1} \eta$) in time interval $[0,t]$.
I would like to show that $$ m_1(t) := N_1(t) - \int_0^t p_1(\eta(s)) ds $$ is a martingale (and similarly for $m_{-1}$) with respect to natural filtration.
Could you give me any hints how I should do this?