# Left continuous adapted process X is optional

I was reading through "Limit Theorems for Stochastic Processes" by Jacod and Shiryaev and came across

1.24 Proposition: Every process $$(X_t)_{t \geq 0}$$ that is left continuous and adapted is optional.

The setup is the follwoning: Let $$(\Omega,\mathcal{F},(F_t)_{t \geq 0}, P)$$ be a filtered probability space with right continuous filtration.

In the proof, they define a new sequence of process $$X^n := \sum_{k \in \mathbb{N}}X_{k/2^n} 1_{[[k/2^n,(k+1)/2^n[[}$$ and claim that for every $$(t,\omega) \in [0,\infty) \times \Omega$$, we have $$X^n_t(\omega) \rightarrow X_t(\omega)$$ for $$n \rightarrow \infty$$. Why does $$X^n$$ converge pointwise to $$X$$ ? I do not see why this has to hold.

Thanks a lot in advance!

## 1 Answer

It's pretty simple: For $$t \geq 0$$, we have $$X_t^n = X_{2^{-n}\lfloor t2^n \rfloor}.$$ Now notice that $$2^{-n}\lfloor t2^n \rfloor \leq t$$ and $$2^{-n}\lfloor t2^n \rfloor \to t$$ ($$n \to \infty$$) and use the left continuity of $$X$$.