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If $X_t$ is right-continuous and adapted process, is $X_{t-}$ always predictable? Or there are other conditions that must be satisfied? I've become a bit unsure due to the following example in "Introduction to Stochastic Calculus with Applications":

Example 8.3: Right-continuous adapted processes may not be predictable, even though they can be approached by left-continuous processes, for example, $X_\epsilon(t)= \lim_{\epsilon\rightarrow 0} > {X((t+\epsilon)−)}$

learningmath
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If $(X_t)_{t\geq 0}$ is right-continuous with finite left limits (i.e. it is càdlàg) and adapted, then $(X_{t-})_{t\geq 0}$ is left-continuous and adapted and hence it is predictable.

I'm pretty sure you need the left limits to be finite in order to justify that $X_{t-}$ is finite.

Stefan Hansen
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