6

Now consider an Ornstein-Uhlenbeck process X = (Xt)t20, defined by the stochastic differential

equation

(2)

dXt = -XXtdt + odZt,

where Z = (Zt)t>o is a standard Brownian motion under the probability measure P and > > 0.

(i) Given a starting value, Xo = 0, solve the stochastic differential equation (2).

(ii) By simulating N paths of the Ornstein-Uhlenbeck process described above, approximate the

transition density of the process and plot it for particular values of A and o. In your code, you

will need to make NV large. How large will be up to you - you will face a trade-off between

running time and accuracy. Compare the transition density of the Ornstein-Uhlenbeck

process with (x, t), the transition density of the standard Brownian motion.

(iii) The exact transition density for the Ornstein-Uhlenbeck process starting at zero is given by

(3)

p(x, t) =

1

V2TU(t)

e 2 v (t ) ,

where

u (t ) =

0 2

27 ( 1 - e-2xt).

Show that (3) satisfies the following partial differential equation

(4)

1 2 22

2 2x2 P(x, t) +. (xp(x, t)) = P(x, t).

(iv) Find lime->op(x, t) and limt->cop(x, t). How would you describe the long-run behavior of

an Ornstein-Uhlenbeck process?