4.4. Argue from basic principles that T 0 W(t,)dW(t,) = 12 W(T,)2T by forming a partition over the time interval [0,T], approximating the

4.4. Argue from basic principles that T

0 W(t,ω)dW(t,ω) = 12



W(T,ω)2−T



by forming a partition over the time interval [0,T], approximating the integral as In(ω) =

n−1 j=0 WjΔWj , and showing that these almost surely tend to I(ω) =

12



W(T,ω)2−T



by considering I−In.

1. Argue that I−In= 12

n−1 j=0



(ΔWj)2−Δtj



by writing 12



W(T,ω)2−T



= 12 n−1 j=0



Δ(W2 j )−Δtj



.

2. Then use some of the ideas appearing in the proof of the Ito isometry and in the steps leading to (2.2) to show that the distance between the integral I(ω)

and the approximate In(ω), measured by E[(I−In)2] , must tend to zero.