Expectation of a product of (many) 1-dimensional Brownian motions.

Question

Let $0=t_0<t_1<t_2<\ldots$ be a sequence of positive reals. Denote by $B(t)$ the 1-dimensional Brownian motion with time $t$. It is easy to show the the expectation of the product of two Brownian motions depends only on the length of the shorter.

Is it true that $\mathbb{E}\left(\prod_{i=1}^n B(t_i)\right)$ is a function of $t_1$ and $n$ only? Phrased differently: is it true that this term does not change if I keep $t_1,n$ but modify $t_i$ for any (or all) $i>1$?

And last thing: does it make sense to speak about the random variable $\prod_{n=0}^\infty B(t_n)$ for certain strictly increasing sequences $\{t_n\}$?

Answer 1

Is it true that $\mathbb{E}\left(\prod_{i=1}^n B(t_i)\right)$ is a function of $t_1$ and $n$ only?

Well, no... $$ E(B(t_1)B(t_2)B(t_3)B(t_4))=2t_1t_2+t_1t_3. $$

does it make sense to speak about the random variable $\prod_{n=0}^\infty B(t_n)$ for certain strictly increasing sequences $\{t_n\}$?

Unless $t_0=0$ or the times $t_n$ are random and chosen very specifically, the infinite product diverges.