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Same Finite-Dimensional Distribution

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Theorem
StochasticDiffs

Let X=(Xt)t≄0,Y=(Yt)t≄0X=(X_{t})_{t\ge 0},Y=(Y_{t})_{t\ge 0} be processes on (Ī©,F,P)(\Omega,\mathcal{F},P).X,YX,Y have the same finite dimensional distribution if āˆ€0≤t1<t2<⋯<tN,Ā āˆ€n∈Nāˆ—\forall 0\le t_{1}<t_{2}<\dots<t_{N}, \ \forall n\in\mathbb{N}^{*}, āˆ€B1,B2,…,BN∈B(R)\forall B_{1},B_{2},\dots,B_{N}\in\mathcal{B}(\mathbb{R}) we have that P(Xt1∈B1,…,XtN∈BN)=P(Yt1∈B1,…,YtN∈BN)P(X_{t_{1}}\in B_{1},\dots,X_{t_{N}}\in B_{N})=P(Y_{t_{1}}\in B_{1},\dots,Y_{t_{N}}\in B_{N})