Hi all, For III), how do you integrate the expression: std normal*cumulative normal from 0 to infinity in the last equation on page 31? For IV), could you help explain further on why the probability is 1? Thanks very much for your help!
Hi For part (iii), the integrand is of the form f'(x)f(x), therefore the integral is 0.5*f(x)^2. The reverse operation is easier to see: differentiate 0.5*f(x)^2 wrt x and you get f'(x)f(x) via the chain rule. For part (iv), if you repeatedly zoom in to the origin of any sample path of standard Brownian motion then eventually you'll see it cross the x-axis. This is because Brownian motion never calms down - it always wobbles - due to the Gaussian increments. In fact, standard Brownian motion returns to zero infinitely often around the origin. To see this you need to keep zooming in.
