# A question about a definition of Brownian motion in a lecture note

Definition 3.2 (Filtration). A discrete filtration of a set $$\Omega$$ is a collection $$\left\{\mathcal{F}_n\right\}$$ of $$\sigma$$-algebras of subsets of $$\Omega$$ such that $$\mathcal{F}_n \subset \mathcal{F}_{n+1}$$ for all $$n \in \mathbb{N}$$. A continuous filtration of a set $$\Omega$$ is a collection $$\left\{\mathcal{F}_n\right\}$$ of $$\sigma$$-algebras of subsets of $$\Omega$$ such that $$\mathcal{F}_s \subset \mathcal{F}_t$$ for all $$s

Definition 3.3 (Natural Filtration). In particular, for a discrete stochastic process $$X_n$$, the natural filtration $$\left\{\mathcal{F}_n\right\}$$ is a filtration such that each $$\mathcal{F}_n$$ is the $$\sigma$$-algebra generated by $$X_1, \ldots, X_n$$. Namely, $$\mathcal{F}_n$$ contains all the information in $$X_1, \ldots, X_n$$

Definition 3.4 (Martingale). A sequence of random variables $$M_0, M_1, \ldots$$ is called a martingale with respect to the filtration $$\left\{\mathcal{F}_n\right\}$$ if:

• (1) For each $$n, M_n$$ is an $$\mathcal{F}_n$$-measurable random variable with $$\mathbb{E}\left[\left|M_n\right|\right]<\infty$$
• (2) If $$m, then $$\mathbb{E}\left[M_n \mid \mathcal{F}_m\right]=M_m$$. Or we can write it in another form: $$\mathbb{E}\left[M_n-M_m \mid \mathcal{F}_m\right]=0 .$$

Moreover, a martingale $$M_t$$ is called continuous martingale if with probability one the function $$t \rightarrow M_t$$ is a continuous function.

Definition 4.1 (Brownian Motion). A stochastic process $$B_t$$ is a Brownian motion with respect to filtration $$\left\{\mathcal{F}_t\right\}$$ if it has the following three properties:

• (1) For $$s, the distribution of $$B_t-B_s$$ is normal with mean 0 and variance $$t-s$$. We denote this by $$B_t-B_s \sim N(0, t-s)$$
• (2) If $$s, the random variable $$B_t-B_s$$ is independent of $$\mathcal{F}_s$$
• (3) With probability one, the function $$f: t \mapsto B_t$$ is a continuous function of t.

Then the author presents a theorem and its proof.

Theorem 4.3. A standard Brownian motion $$B_t$$ is a continuous martingale with respect to filtration $$\left\{\mathcal{F}_t\right\}$$

Proof. Let $$s. Using properties of conditional expectation, we have that $$\mathbb{E}\left[B_t \mid \mathcal{F}_s\right] = \mathbb{E}\left[B_s \mid \mathcal{F}_s\right]+\mathbb{E}\left[B_t-B_s \mid \mathcal{F}_s\right]=B_s+\mathbb{E}\left[B_t-B_s\right]=B_s .$$ By the third property of Brownian motion, we have proved $$B_t$$ is a continous martingale.

I think for $$\mathbb{E}\left[B_s \mid \mathcal{F}_s\right] = B_s$$ to hold, we need $$B_s$$ is $$\mathcal{F}_s$$-measurable. However, I could not find this condition in the Def 4.1.

Could you confirm that Def 4.1 lacks the condition that $$(B_t)$$ is adapted to the filtration $$(\mathcal{F}_t)$$?

Some details are missing from the notes. A filtration $$(\mathscr{F}_t)_{t\geq0}$$ is called admissible if $$\sigma(B_s,s\leq t)=:\mathscr{F}_t^B\subseteq \mathscr{F}_t,\,\forall t\geq 0$$ and $$B_t-B_s\perp \mathscr{F}_s$$ for $$0\leq s\leq t$$. Then, given a Brownian motion $$(B_t)_{t \geq 0}$$, we have that $$B_t$$ is a $$\mathscr{F}_t$$-martingale if $$(\mathscr{F}_t)_{t\geq0}$$ is admissible. Indeed: $$E[B_t-B_s|\mathscr{F}_s]=E[B_t-B_s]=0$$ and $$E[B_s|\mathscr{F}_s]=B_s$$ because $$B_s$$ is $$\mathscr{F}_s$$-measurable by admissibility.