Need help for Ito Isometry proof

[Sa7oru]

Hi,

There is one step in Ito Isometry proof I don't understand.

B_t(w) represents the position of w at time t, then E[(B_tj+1(w)-B_tj(w))²] = t_j+1 - t_j.

Why is the expectation of the square of the difference of 2 positions equal to their time difference?

Any hint, please.

Thank you.

 

[Eynstone]

Could you clarify what Bt(w) is & whether it's normally distributed?

 

[snipez90]

I'm guessing B_t(w) is Brownian motion, and yes it's normally distributed because its increments, B_t(w) - B_s(w) are normally distributed (~ N(0, t-s)) if s < t. Indeed, this is the precise property that gives the result the OP wants to understand.