6. (20 points) Consider the stochastic differential equation
dX? = ?(? - X?)dt + ?dW?, X? = x?,
(0.1)
where ? > 0, x?, ? ? ? and ? > 0 are constants. In this problem we solve this SDE, i.e., we
find a process which satisfies the stated dynamics. The solution is called the Ornstein-Uhlenbeck
process which is applied in interest rate models. [It may also be regarded as the continuous
time analogue of the AR(1) time series model.]
(a) Suppose X? satisfies (0.1). Apply Itô's formula to the process Y? = e^{?t}X?. Show that
Y? = Y? + \int_0^t ??e^{?s}ds + \int_0^t ?e^{?s}dW?.
(b) Note that the integrands above are all deterministic. Thus show that
X? = x?e^{-?t} + ?(1 - e^{-?t}) + ? \int_0^t e^{-?(t-s)}dW?.
This gives an explicit solution to the SDE.
(c) Since the integrands are all deterministic, {X?} is actually a Gaussian process. Find the
distribution of X? at a given time t > 0. Formally, what is the limiting distribution when
t ? ?? (It is a normal distribution.)