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Assume (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0}(Ft)t≥0 satisfies the Filtration. Let (Xt)t≥0(X_{t})_{t\ge 0}(Xt)t≥0 be a (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0}(Ft)t≥0-martingale then (Xt)t≥0(X_{t})_{t\ge 0}(Xt)t≥0 admits a Càdlàg version. i.e. ∃(X~t)t≥0(Ft)t≥0-adapted ∀t≥0 where Xt=X~t a.s.\exists(\tilde{X}_{t})_{t\ge 0} (\mathcal{F}_{t})_{t\ge 0}\text{-adapted }\forall t\ge 0\text{ where }X_{t}=\tilde{X}_{t}\text{ a.s.}∃(X~t)t≥0(Ft)t≥0-adapted ∀t≥0 where Xt=X~t a.s.
We see that the X~n\tilde{X}_{n}X~n preserves XnX_{n}Xn’s Martingale properties: ∀0≤s<t:X~s=Xs=E[Xt∣Fs]=E[X~t∣Fs] a.s.\forall {0}\le s<t:\tilde{X}_{s}=X_{s}=E[X_{t}|\mathcal{F}_{s}]=E[\tilde{X}_{t}|\mathcal{F}_{s}]\text{ a.s.}∀0≤s<t:X~s=Xs=E[Xt∣Fs]=E[X~t∣Fs] a.s.