1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ By Kolmogorov extension theorem (see here, here, here or theorem 2.2 of chapter 2 (p. 50) of Karatzas/Shreve). $\endgroup$ – Sayantan Apr 20 at 3:52
  • $\begingroup$ The question does not make sense unless you define the term distribution of a stochastic process''. One can talk about the distribution if BM as a measure on $C[0,1]$ or on $\mathbb R ^{[0,1]}$ with the product topology. $\endgroup$ – Kavi Rama Murthy Apr 20 at 5:50
0
$\begingroup$

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 ${\mathcal C}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.