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I have a question on distributions for some stoachastic processes.

Let $(\Omega_i,\mathcal{F_i},P_i)$, $i=1,2$, be probability spaces. Let $X=(X_t)_{t \in [0,\infty)}$ be an $\mathbb{R}^d$-valued stochastictic process defined on the probability space $(\Omega_1,\mathcal{F_1},P_1)$. The sample path of $X$ is continuous. Let $B=(B_t)_{t \in [0,\infty)}$ be a $d$-dimensional Brownian motion on $(\Omega_2,\mathcal{F_2},P_2)$.

We assume that for every $t \ge 0$ and bounded continuous function $f \colon \mathbb{R}^d \to \mathbb{R}$, \begin{align*} E_1[f(X_t)]=E_{2}[f(B_t)] \end{align*} Here, $E_1$ and $E_2$ denote the expectations under $P_1$ and $P_2$, respectively.

Then, can we show that the finite dimensional distributions of $X$ and $B$ are the same ? That is, for any $m \in \mathbb{N}$, $0 \le t_1 \le t_2\le \cdots \le t_m$, and any Borel subsets $A_1,A_2,\ldots, A_m$, \begin{align*} &P_1[X_{t_1} \in A_1,X_{t_2} \in A_2,\ldots, X_{t_m} \in A_m] \\ &=P_2[B_{t_1} \in A_1,B_{t_2} \in A_2,\ldots, B_{t_m} \in A_m]. \end{align*}

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Take $X_t=\sqrt{t}B_1$. Then at any fixed point in time $t$, $X_t=B_t$ in distribution (normal distribution with mean 0 and variance $t$), but as processes $X\neq B$ in distribution since $X$ has no independent increments.

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  • $\begingroup$ You should take $X_t=\sqrt t B_1$. $\endgroup$ Commented Sep 23, 2021 at 23:13

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