Simple question about the definition of Brownian motion

I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition:

$\forall 0\leq t_0<t_1<…<t_m: B_{t_1}-B_{t_0},…,B_{t_m}-B_{t_{m-1}}$ are independent random variables.

But today I come across with formally another definition of Brownian motion! Instead of the above condition the book demands the following property:

$\forall t,h\geq 0$, $B_{t+h}-B_t$ is independent of $\lbrace B_u : 0\leq u\leq t \rbrace$.

Now I’m wondering whether both properties are indeed equivalent or not?! Hopefully someone can help me.

Solutions Collecting From Web of "Simple question about the definition of Brownian motion"

Yes, they are indeed equivalent.

Proof: Let $X_1,\ldots,X_n$ arbitrary random variables and denote by

$$S_j := \sum_{k=1}^j X_k$$

the $j$-th partial sum. Since $$X_j = S_j – S_{j-1} \qquad \quad S_j = \sum_{k=1}^j X_k$$

it is not difficult to see that

$$\sigma(X_1,\ldots,X_n) = \sigma(S_1,\ldots,S_n).$$

If we set $X_j := B_{t_j}-B_{t_{j-1}}$ for a Brownian motion $(B_t)_{t \geq 0}$, then this shows

$$\sigma(B_{t_1},\ldots,B_{t_n}) = \sigma(B_{t_1}-B_{t_0},\ldots,B_{t_n}-B_{t_{n-1}}). \tag{1}$$

Consequently,

$$\begin{align*} \sigma\{B_u; u \leq t\} &\stackrel{\text{Def}}{=} \sigma \left( \bigcup_{\substack{t_1<\ldots<t_n \leq t \\n \in \mathbb{N}}} \sigma(B_{t_1},\ldots,B_{t_n}) \right) \\ &\stackrel{(1)}{=} \sigma \bigg( \underbrace{\bigcup_{\substack{t_1<\ldots<t_n \leq t \\n \in \mathbb{N}}} \sigma(B_{t_1}-B_{t_0},\ldots,B_{t_n}-B_{t_{n-1}})}_{=:\mathcal{G}} \bigg). \end{align*}$$

Since $\mathcal{G}$ is a $\cap$-stable generator of $\sigma(B_u; u \leq t)$, we conclude that $B_{t+u}-B_t$ is independent from $\sigma(B_u; u \leq t)$ if, and only if, $B_{t+h} -B_t$ is independent from $\mathcal{G}$. And this is equivalent to $B_{t+h}-B_t$ being independent from $B_{t_n}-B_{t_{n-1}},\ldots,B_{t_1}-B_{t_0}$ for all $t_1 < \ldots < t_n \leq t$.

Literature: See e.g. René L. Schilling/Lothar Partzsch: Brownian Motion – An Introduction to Stochastic Processes (Lemma 2.14).