Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel-$\sigma$-algebra $\mathcal{E}$ and we assume that $t\rightarrow E_{X_{t}}f(X_{s})$ is right continuous everywhere for each bounded continuous function f.
For $x\in E$ define $\sigma_{x}=\inf\{t>0\mid X_{t}\neq x\}$.
I've come to the conclusion that $\sigma_{x}$ is an optional time. I would now like to use the Markov property of $X$ to show that for every $x\in E$ it holds that
$$P_{x}\{\sigma_{x}>t+s\}=P_{x}\{\sigma_{x}>t\}P_{x}\{\sigma_{x}>s\}$$ for all $s,t\geq 0$
I've tried using the indicator function $\mathbb{1}_{\sigma_{x}>t+s}$ but didnt come to the desired result. Could anyone help me prove this?
Indeed $\sigma_x$ is exponentially distributed under $P_x$. To see this, note that $[\sigma_x\gt s]\subseteq[X_s=x]$ hence $$ P_x[\sigma_x\gt t+s\mid\sigma_x\gt s]=P_x[\sigma_x\gt t+s\mid X_s=x]=P_x[\sigma_x\gt t]. $$ This is specific to the starting distribution $P_x$. For other starting distributions, the distribution of $\sigma_x$ is the barycenter of the exponential distribution $P_x\circ\sigma_x^{-1}$ and of the Dirac mass at $0$ with respective weights $P[X_0=x]$ and $1-P[X_0=x]$.