Math 236, Stochastic Differential Equations
Problem set 4
February 17, 2009
These problems are due on
Tuesday February 24
. You can give them to me in class, drop
them in my office, or put them in my mailbox outside the mathematics department.
Problem 1
Let
X
(
t
) = (
X
i
(
t
)) be the
n
vector valued OrnsteinUhlenbeck process. We use arguments for
the time since we want to use subscripts for the components of the vector. Its SDE is
dX
(
t
) =
AX
(
t
)
dt
+
σdB
(
t
)
,
X
(0) =
x
where
A
and
σ
are
n
×
n
matrices and
B
(
t
) = (
B
i
(
t
)) is a vector of
n
independent standard
Brownian motions. Show (i) that the covariance matrix
C
(
t
) =
E
x
{
X
(
t
)
X
(
t
)
T
}
satisfies the matrix
differential equation (Lyapounov equation)
dC
(
t
)
dt
=
AC
(
t
) +
C
(
t
)
A
T
+
σσ
T
,
C
(0) =
xx
T
Here the superscript
T
denotes transpose. Find (ii) an integral representation for the covariance
matrix
C
(
t
). Show (iii) that if all the eigenvalues of
A
have negative real parts then the equilibrium
(time homogeneous) OU process has the form
X
e
(
t
) =
integraldisplay
t
∞
e
A
(
t

s
)
σdB
s
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 Winter '08
 Papanicolaou
 Differential Equations, Equations, Brownian Motion, Boundary value problem, Covariance matrix, negative real parts, matrix diﬀerential equation

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