I have a rightcontinuous adapted $\bar{\mathbb{R}}$-valued stochastic process $(X_t)_{t\geq 0}$ with $X_s=\infty$ implies $X_t=\infty$ for $t>s \geq 0$. Since $\bar{\mathbb{R}}$ is homeomorph to $[-1,1]$ it makes sense to talk about rightcontinuity for the process $X$ that may also take the value $\infty$ or $-\infty$.
Is it true that $X$ is an optional process, i.e. is it measureable w.r.t. the optional $\sigma$-algebra? As $X$ can take the value $\infty$ I don't know how to check this. I would be grateful for hints or a solution.
If $f$ is a homeomorphism of $\bar{\Bbb R}$ onto $[-1,1]$ and $Y_t:=f(X_t)$, then $Y$ is real-valued, right-continuous and adapted, hence optional provided the filtration is right-continuous.