$(20$ marks$)($Properties of Brownian Motion$)$

Let

$Z_i-=Z(t_{i+1})-Z(t_i)=,(t_i)\sqrt[\phantom{2}]{t}$

$(t_i),,N(0,1)$

$Z(0),=,0$

and

$I(t)={\displaystyle \underset{i=0}{\overset{i=K-1}{}}}(Z_i{)}^2$

$t=\frac{T}{K}$

Write matlab code to evaluate $I(t).$ For $T=1,$ compute the value of $I(t).$ Use $100000$

paths, and compute the mean and variance of the result, for $K=100,200,400,800,1600$

timesteps. Vectorize your code so that there is only one explicit for loop over the timesteps.

Plot a graph of var$(I(t))$ versus $t.$ Explain what you see. Relate this to a crucial step in

deriving Ito's Lemma.