The setup of my question is the following: Suppose that we have a measurable space $(\Omega,\mathcal{F})$ and a filtration $\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$ on it. Let $\mathcal{P}(\mathbf{F})$ be the predictable $\sigma$-algebra, that is, the $\sigma$-algebra generated by all real-valued left-continuous processes adapted to the filtration $\mathbf{F}$. Let $\mathcal{O}(\mathbf{F})$ be the optional $\sigma$-algebra, that is, the $\sigma$-algebra generated by all real-valued right-continuous process with left-limits adapted to the filtration $\mathbf{F}$.

I was trying to find conditions on the underlying measurable space $(\Omega,\mathcal{F})$ or the filtration $\mathbf{F}$, under which the predicatble $\sigma$-algebra $\mathcal{P}(\mathbf{F})$ coincides with the optional $\sigma$-algebra $\mathcal{O}(\mathbf{F})$. For example, if the filtered space $(\Omega,\mathcal{F})$ supports a process $X=(X_t)_{t \geq 0}$ with values in a metric space $(E,d)$ for example, such that the paths $\mathbb{R}_+ \ni t \mapsto X_t \in E$ are continuous, and the filtration $\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$ is generated by $X$, that is, $$\mathcal{F}_t = \sigma(X_r : 0 \leq r \leq t),$$ does the equality $$\mathcal{P}(\mathbf{F}) = \mathcal{O}(\mathbf{F})$$ hold?

If this condition is too weak, does there exist some other general condition under which the predictable $\sigma$-algebra coincides with the optional $\sigma$-algebra?