For $i=1,2$, let $\{X_t^i\}_{t\geq 0}$ be $\Bbb R^d$-valued stochastic processes adapted to $\{\mathscr F^i_t\}_{t\geq 0}$ on the probability space $(\Omega^i,\mathscr F^i,\Bbb P^i)$. Suppose the two processes have the same finite dimensional distributions, i.e. for any $n\in\Bbb N$ and $t_1<\cdots<t_n$ and $A\in \mathscr B(\Bbb R^{dn})$ we have
$$\Bbb P^1[(X^1_{t_1},\cdots,X^1_{t_n})\in A]=\Bbb P^2[(X^2_{t_1},\cdots,X^2_{t_n})\in A]$$
Is it true that $\{X_t^1\}_{t\geq 0}$ under $\Bbb P^1$ has the same law as $\{X_t^2\}_{t\geq 0}$ under $\Bbb P^2$?
In the context where the $X^i$ are solutions to a SDE, the book Brownian Motion and Stochastic Calculus (Chapter 5, Proposition 3.10) proves the equivalence in law by showing that the finite dimensional distributions are the same. However, it is not clear to me how the equivalence in law follows.
Best Answer
I understand the question is that is two process have the same "finite dimensional distribution" then they are the same. Notice a stochastic process induces a measure on ${\mathbb R}^{[0, \infty)}$, the space of all maps $[0, \infty)\to {\mathbb R}$, thus we just need to show $P_1=P_2$, where $P_i$ is the measure on ${\mathbb R}^{[0, \infty)}$ induced by $X^i$, where $i=1, 2$.
This can be done by "Dynkin $\pi$-$\lambda$ theorem", see p49 of the book you mentioned, or see https://en.wikipedia.org/wiki/Dynkin_system
Briefly, consider the collection ${\mathcal F}$ of subsets $A\in{\mathbb R}^{[0, \infty)}$ so that $P_1(A)=P_2(A)$. Now ${\mathcal F}$ contains the collection ${\mathcal C}$ of all "cylinder sets", i.e. those of the form $\{X\,|\,(X_{t_1}, ...,X_{t_n})\in B\}$, where $B$ is Borel in ${\mathbb R}^n$. The collection ${\mathcal C}$ forms a "$\pi$-system", i.e. if $A, B\in {\mathcal C}$ then $A\cap B\in {\mathcal C}$. Then notice ${\mathcal F}$ is a "Dynkin system", i.e. if $A_1,A_2, ...$ is a disjoint sequence in ${\mathcal F}$, then $\cup_i A_i\in{\mathcal F}$, and if $A\in {\mathcal F}$ then ${\mathbb R}^{[0, \infty)}-A\in {\mathcal F}$. Now apply the Dynkin π-λ theorem, we see ${\mathcal F}$ contains the $\sigma$-algebra generated by ${\mathcal C}$, and this says $P_1=P_2$ since the canonical $\sigma$-algebra on ${\mathbb R}^{[0, \infty)}$ is just the one generated by ${\mathcal C}$.