(Adapted from Bjrk (2009)) Define the process y = (yt) by yt = eazt , where a...

(Adapted from Björk (2009)) Define the process y = (yt) by yt = eazt , where a is a constant and z = (zt) is a standard Brownian motion. Find the dynamics of y. Show that yt = 1 +

1 2

a2 t

0 ys ds + a t

0 ys dzs.

Define m(t) = E[yt]. Show that m satisfies the ordinary differential equation m

(t) = 1 2

a2m(t), m(0) = 1.

Show that m(t) = ea2 t/2 and conclude that E



eazt

= ea2 t/2.