Brownian motion in discrete time

Now suppose that we have a collection of $n$ independent standard normal variables defined on a probability triple $\left( \Omega ,\mathscr{F}%% ,P\right)$. We call $P$ the market [probability] measure, and we denote this sequence of random variables with $\left\{ Y_{k}\right\} _{k=1}^{n}$ We define the discrete time Brownian motion to be

$$B_{0} =$$ 0

$$B_{k} = \sum_{j=1}^{k}Y_{j},\forall k=1,\ldots ,n$$

Apparently if we know the values of $\left\{ Y_{j}\right\} _{j=1}^{k}$ then we know the values of $\left\{ B_{j}\right\} _{j=1}^{k}$, and inversely. We now define the filtration [just an increasing sequence of $\sigma$-algebras] in the following way

$$\mathscr{F}_{0} = \left\{ \emptyset ,\Omega \right\}$$

$$\mathscr{F}_{k} = \sigma \left( Y_{1},\ldots ,Y_{k}\right)$$

$$= \sigma \left( B_{1},\ldots ,B_{k}\right) ,\forall k=1,\ldots ,n$$

These $\sigma$-algebras will keep track of the information someone has, just by observing the Brownian motion behavior. The relation between $%% \left\{ Y_{j}\right\} _{j=1}^{k}$ and $\left\{ B_{j}\right\} _{j=1}^{k}$ is illustrated in figure 7.1.

Figure 6.1: Discrete time asset price and the generating Brownian motion