predictable process in continuous times.

Question

If $X_t$ is a stochastic process in $(\Omega,\mathcal F,\mathcal F_t)$ which satisfy usual conditions.

Can we get $X_t$ is a predictable process if $X_t$ is adapted to $\mathcal F_{t-}$?

since predictable process in continuous time is defined by the predictable $\sigma$-algebra which is generated by the left continuous process. But I don't know the path properties of $X_t$.

Answer 1

No, a stochastic process $X=(X_t)_{t \geq 0}$ which is adapted to $\mathcal{F}_{t-}$ for each $t \geq 0$ is, in general, not predictable.

Example: Let $A \subseteq [0,\infty)$ be a set which is not Borel-measurable and define

$$X_t(\omega) := \begin{cases} 1, & t \in A, \\ 0, & t \notin A. \end{cases}$$

Since $\omega \mapsto X_t(\omega)$ is a constant function for each fixed $t \geq 0$, it is $\mathcal{F}_{t-}$-adapted (in fact, it is adapted with respect to any filtration). However, since

$$X^{-1}(\{1\}) = \{(t,\omega) \in [0,\infty) \times \Omega; X_t(\omega)=1\} = A \times \Omega$$

is not contained in the predictable $\sigma$-algebra (as $A$ is not measurable), the process $X$ is not predictable.