A process is Càdlàg (i.e. right continuous with left limits) if ∀ω∈Ω:t↦Xt(ω) is right continuous and has left limits everywherei.e.
the left limit f(t−):=s↑t−limf(s) exists and;
the right limit f(t+):=s↓t+limf(s)exists and equals f(t).
Intuition
Why right continuity? Well, here we have a great example of why: i.e. Right continuity allows us to use the rationals when indexing time. This then allows us to use the countable properties of measure theory to do various things.
Example
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Theorem (Existence of Càdlàg Version)
Assume (Ft)t≥0 satisfies the Usual conditions. Let (Xt)t≥0 be a (Ft)t≥0-martingale then (Xt)t≥0 admits a Càdlàgversion. i.e. ∃(X~t)t≥0(Ft)t≥0-adapted ∀t≥0 where Xt=X~t a.s.
Remark
We see that the X~n preserves Xn’s Martingale properties: ∀0≤s<t:X~s=Xs=E[Xt∣Fs]=E[X~t∣Fs] a.s.
i.e. Right continuity allows us to use the rationals when indexing time. This then allows us to use the countable properties of measure theory to do various things.