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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

$\displaystyle B_{0}$ $\displaystyle =$ 0  
$\displaystyle B_{k}$ $\displaystyle =$ $\displaystyle \sum_{j=1}^{k}Y_{j},\forall k=1,\ldots ,n$.  

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
$\displaystyle \mathscr{F}_{0}$ $\displaystyle =$ $\displaystyle \left\{ \emptyset ,\Omega \right\}$  
$\displaystyle \mathscr{F}_{k}$ $\displaystyle =$ $\displaystyle \sigma \left( Y_{1},\ldots ,Y_{k}\right)$  
  $\displaystyle =$ $\displaystyle \sigma \left( B_{1},\ldots ,B_{k}\right) ,\forall k=1,\ldots ,n$.  

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
\begin{figure}\centering\begin{texdraw} \drawdim cm %% \linewd 0.05 %% \move(1 1... ...\htext(8.5 6.1){$B_4$} %% \end{texdraw}<tex2html_comment_mark>801%% \end{figure}

next up previous contents
Next: The market Up: Modelling: Continuous state space Previous: The building blocks   Contents
Kyriakos 2003-03-17