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Let $C([0,\infty), R)$ be the canonical space of continuous functions. Assume $(\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\geq 0})$ is a measurable space with a filtration. Let $P, Q$ be two probability measures on $(\Omega, \mathcal{F})$. Assume $X_{t}$ and $Y_{t}$ are two stochastic processes adapted to $\{\mathcal{F}_{t}\}_{t\geq 0}$. If for any borel set $A$ and $t$, $$ P(X_{t}\in A)= Q(Y_{t}\in A) $$ Can we conclude that the law on $C([0,\infty), R)$ induced by $(X_{t}, P)$ is the same as that of $(Y_{t}, Q)$? Any references are very appreciated.

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  • $\begingroup$ If it is not true. How about one of them is a Levy process. $\endgroup$ – Hengyu Mar 19 '14 at 5:12
  • $\begingroup$ Just for Discussion. To strengthen the condition. If at the beginning, we require that\\$$ X_{t} = Y_{t} $$ and $$ P(X_{T} \in A ) = Q(X_{t}\in A )$$ Is $$ law(X_{t}|P) = law(X_{t}|Q)$$. Here $P$ are $Q$ are mutually absolutely continuous with each other. $\endgroup$ – Hengyu Mar 19 '14 at 17:11
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No. Try $(X_t)$ standard Brownian motion and $Y_t=\sqrt{t}\cdot Y_1$ for every $t$, where $Y_1$ is standard normal.

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If only the marginals match it is not true, here is a nice counter example the fake Brownian Motion

Best regards

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  • $\begingroup$ Being a modification of each other is stronger than requiring the finite dimensional distributions to be the same. $\endgroup$ – Stefan Hansen Mar 19 '14 at 9:34
  • $\begingroup$ My memory has leaked I was pretty sure I had read something like that in the reference I mentioned but I wasn't able to find it so I edit my answer thanks for pointing that out Regards. $\endgroup$ – TheBridge Mar 19 '14 at 9:54
  • $\begingroup$ That's ok. Thank you very much. I think when $(N_{t}, P)$ is a Levy process, the conclusion is true. The proof is as follows. If $(N_{t}, P)$ is a Levy process. Then its distribution of $(N_{t}, P)$ is infinitesimal divisible. We observe that the distribution of $(N_{t}, Q)$ is also infinitesimal divisible. Since there is a one-to-one correspondence of Levy process and infinitesimal divisible distribution( Sato's book: Levy process and infinitesimal distribution). Then we conclude that $law(N_{t}, P)=law(N_{t}, Q)$. $\endgroup$ – Hengyu Mar 19 '14 at 17:09

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