4Stochastic differential equations
III Stochastic Calculus and Applications
4.2 Examples of stochastic differential equations
Example
(The Ornstein–Uhlenbeck process)
.
Let
λ >
0. Then the Ornstein–
Uhlenbeck process is the solution to
dX
t
= −λX
t
dt + dB
t
.
The solution exists by the previous theorem, but we can also explicitly find one.
By Itˆo’s formula applied to e
λt
X
t
, we get
d(e
λt
X
t
) = e
λt
dX
t
+ λe
λt
X
t
dt = dB
t
.
So we find that
X
t
= e
−λt
X
0
+
Z
t
0
e
−λ(t−s)
dB
s
.
Observe that the integrand is deterministic. So we can in fact interpret this as
an Wiener integral.
Fact.
If
X
0
=
x ∈ R
is fixed, then (
X
t
) is a Gaussian process, i.e. (
X
t
i
)
n
i=1
is
jointly Gaussian for all
t
1
< ··· < t
n
. Any Gaussian process is determined by
the mean and covariance, and in this case, we have
EX
t
= e
−λt
x, cov(X
t
, X
s
) =
1
2λ
e
−λ|t−s|
− e
−λ|t+s|
Proof. We only have to compute the covariance. By the Itˆo isometry, we have
E((X
t
− EX
t
)(X
s
− EX
s
)) = E
Z
t
0
e
−λ(t−u)
dB
u
Z
s
0
e
−λ(s−u)
dB
u
= e
−λ(t+s)
Z
t∧s
0
e
λu
du.
In particular,
X
t
∼ N
e
−λt
x,
1 − e
−2λt
2λ
→ N
0,
1
2λ
.
Fact.
If
X
0
∼ N
(0
,
1
2λ
), then (
X
t
) is a centered Gaussian process with stationary
covariance, i.e. the covariance depends only on time differences:
cov(X
t
, X
s
) =
1
2λ
e
−λ|t−s|
.
The difference is that in the deterministic case, the
EX
t
cancels the first
e
−λt
X
0
term, while in the non-deterministic case, it doesn’t.
This is a very nice example where we can explicitly understand the long-time
behaviour of the SDE. In general, this is non-trivial.
Dyson Brownian motion
Let
H
N
be an inner product space of real symmetric
N ×N
matrices with inner
product
N Tr
(
HK
) for
H, K ∈ H
N
. Let
H
1
, . . . , H
dim(H
N
)
be an orthonormal
basis for H
N
.
Definition
(Gaussian orthogonal ensemble)
.
The Gaussian Orthogonal Ensem-
ble GOE
N
is the standard Gaussian measure on H
N
, i.e. H ∼ GOE
N
if
H =
dim H
n
X
r=1
H
i
X
i
where each X
i
are iid standard normals.
We now replace each
X
i
by a Ornstein–Uhlenbeck process with
λ
=
1
2
. Then
GOE
N
is invariant under the process.
Theorem. The eigenvalues λ
1
(t) ≤ ··· ≤ λ
N
(t) satisfies
dλ
i
t
=
−
λ
i
2
+
1
N
X
j6=i
1
λ
i
− λ
j
dt +
r
2
Nβ
dB
i
.
Here
β
= 1, but if we replace symmetric matrices by Hermitian ones, we get
β = 2; if we replace symmetric matrices by symplectic ones, we get β = 4.
This follows from Itˆo’s formula and formulas for derivatives of eigenvalues.
Example
(Geometric Brownian motion)
.
Fix
σ >
0 and
t ∈ R
. Then geometric
Brownian motion is given by
dX
t
= σX
t
dB
t
+ rX
t
dt.
We apply Itˆo’s formula to log X
t
to find that
X
t
= X
0
exp
σB
t
+
r −
σ
2
2
t
.
Example
(Bessel process)
.
Let
B
= (
B
1
, . . . , B
d
) be a
d
-dimensional Brownian
motion. Then
X
t
= |B
t
|
satisfies the stochastic differential equation
dX
t
=
d − 1
2X
t
dt + dB
t
if t < inf{t ≥ 0 : X
t
= 0}.