Exercise . (Adapted from Bjrk ().) Define the process y = (yt) by yt = eazt ,

Exercise . (Adapted from Björk ().) 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 =  +





a

t



ys ds + a t



ys dzs.

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

(t) = 



am(t), m() = .

Show that m(t) = ea t/ and conclude that E



eazt

= ea t/.