I have the following questions: Let $Z$ be a continuous one-dimensional Markov process on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_t = \sigma(Z_s,s \leq t)$. Then show that for all $T > 0$:

for all $t \leq T$ there exists a function $F(t,T,\cdot)$ such that

$$E[e^{-\int_{t}^T R(Z_s) ds} | \mathcal{F}_t] = F(t,T,Z_t),$$

where $R$ is some function. If $Z$ is time-homogeneous then $\forall t\leq T$ there exists a function $G$ such that $$F(t,T,z) = G(T-t,z).$$

So for question (1) I intuitively know that $E[e^{-\int_{t}^T R(Z_s) ds} | \mathcal{F}_t] = E[e^{-\int_{t}^T R(Z_s) ds} | Z_t] $ but how do I proof it? For question (2) I was thinking about Expressing it in terms of $Q_{T-t}f(Z_t)$ where $(Q_{t})_{t \geq 0}$ is the transition semi group, but since there is an integral expression I have no clue how to do this.

I have the following questions: Let $Z$ be a continuous one-dimensional Markov process on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_t = \sigma(Z_s,s \leq t)$. Then show that for all $T > 0$:

for all $t \leq T$ there exists a function $F(t,T,\cdot)$ such that

$$E[e^{-\int_{t}^T R(Z_s) ds} | \mathcal{F}_t] = F(t,T,Z_t),$$

where $R$ is some function. If $Z$ is time-homogeneous then for all $t\leq T$ there exists a function $G$ such that $$F(t,T,z) = G(T-t,z).$$

So for question (1) I intuitively know that $E[e^{-\int_{t}^T R(Z_s) ds} | \mathcal{F}_t] = E[e^{-\int_{t}^T R(Z_s) ds} | Z_t] $ but how do I prove it? For question (2) I was thinking about expressing it in terms of $Q_{T-t}f(Z_t)$ where $(Q_{t})_{t \geq 0}$ is the transition semi-group, but since there is an integral expression I have no clue how to do this.