(Ito's Lemma / Mean-Reversion Model - THIS CAN BE DIFFICULT) Let us define X(t)_0^texp(s)dz(s) where z(s) is a Brownian motion; _0^tdz(s) is an Ito integration; >0 is a constant. As we learned in our class, its short-hand notation is dX(t)exp(t)dz(t) . Let us define Y(t)Y +(Y(0)-Y ) exp(-t)+ exp(-t)X(t) where Y and >0 are constant. Now use Ito's lemma to show that dY(t)=(Y -Y(t))dt+ dz(t) which is often called a Ornstein-Uhlenbeck process. [Hint #: Let f(t,X(t)) Y +(Y(0)-Y ) exp(-t)+ exp(-t)X(t)=Y(t) and apply the Ito's lemma to f(t,X(t)).] An Ornstein-Uhlenbeck process is known to be very useful for modeling a random price Y(t) which tends to revert to the long-term mean Y . Why do you think Y(t) tends to revert to the long-term mean Y in a Ornstein-Uhlenbeck process? Please, give your verbal answer as if you are explaining it into your colleague who does not know much mathematical finance