In mathematics, the Ornstein-Uhlenbeck process (named after Leonard Salomon Ornstein and George Eugene Uhlenbeck ), also known as the mean-reverting process, is a stochastic process given by the following stochastic differential equation
d
r
t
=
−
θ
(
r
t
−
μ
)
d
t
+
σ
d
W
t
,
{\displaystyle dr_{t}=-\theta (r_{t}-\mu )\,dt+\sigma \,dW_{t},\,}
where θ, μ and σ are parameters and W t Wiener process .
three sample paths of different OU-processes with θ=1, μ=1.2, σ=0.3:navy olive red
Solution
This equation is solved by variation of parameters . Apply Itō's lemma to the function
f
(
r
t
,
t
)
=
r
t
e
θ
t
{\displaystyle f(r_{t},t)=r_{t}e^{\theta t}}
d
f
(
r
t
,
t
)
=
θ
r
t
e
θ
t
d
t
+
e
θ
t
d
r
t
{\displaystyle df(r_{t},t)=\theta r_{t}e^{\theta t}\,dt+e^{\theta t}\,dr_{t}\,}
=
e
θ
t
θ
μ
d
t
+
σ
e
θ
t
d
W
t
.
{\displaystyle =e^{\theta t}\theta \mu \,dt+\sigma e^{\theta t}\,dW_{t}.\,}
Integrating from 0 to t we get
r
t
e
θ
t
=
r
0
+
∫
0
t
e
θ
s
θ
μ
d
s
+
∫
0
t
σ
e
θ
s
d
W
s
{\displaystyle r_{t}e^{\theta t}=r_{0}+\int _{0}^{t}e^{\theta s}\theta \mu \,ds+\int _{0}^{t}\sigma e^{\theta s}\,dW_{s}\,}
whereupon we see
r
t
=
r
0
e
−
θ
t
+
μ
(
1
−
e
−
θ
t
)
+
∫
0
t
σ
e
θ
(
s
−
t
)
d
W
s
.
{\displaystyle r_{t}=r_{0}e^{-\theta t}+\mu (1-e^{-\theta t})+\int _{0}^{t}\sigma e^{\theta (s-t)}\,dW_{s}.\,}
Thus, the first moment is given by (assuming that
r
0
{\displaystyle r_{0}}
E
(
r
t
)
=
r
0
e
−
θ
t
+
μ
(
1
−
e
−
θ
t
)
.
{\displaystyle E(r_{t})=r_{0}e^{-\theta t}+\mu (1-e^{-\theta t}).}
Denote
s
∧
t
=
min
(
s
,
t
)
{\displaystyle s\wedge t=\min(s,t)}
Itō isometry to calculate the covariance function by
cov
(
r
s
,
r
t
)
=
E
[
(
r
s
−
E
[
r
s
]
)
(
r
t
−
E
[
r
t
]
)
]
{\displaystyle \operatorname {cov} (r_{s},r_{t})=E[(r_{s}-E[r_{s}])(r_{t}-E[r_{t}])]}
=
E
[
∫
0
s
σ
e
θ
(
u
−
s
)
d
W
u
∫
0
t
σ
e
θ
(
v
−
t
)
d
W
v
]
{\displaystyle =E[\int _{0}^{s}\sigma e^{\theta (u-s)}\,dW_{u}\int _{0}^{t}\sigma e^{\theta (v-t)}\,dW_{v}]}
=
σ
2
e
−
θ
(
s
+
t
)
E
[
∫
0
s
e
θ
u
d
W
u
∫
0
t
e
θ
v
d
W
v
]
{\displaystyle =\sigma ^{2}e^{-\theta (s+t)}E[\int _{0}^{s}e^{\theta u}\,dW_{u}\int _{0}^{t}e^{\theta v}\,dW_{v}]}
=
σ
2
2
θ
e
−
θ
(
s
+
t
)
(
e
2
θ
(
s
∧
t
)
−
1
)
.
{\displaystyle ={\frac {\sigma ^{2}}{2\theta }}\,e^{-\theta (s+t)}(e^{2\theta (s\wedge t)}-1).\,}
It is also possible (and often convenient) to represent
r
t
{\displaystyle r_{t}}
r
t
=
μ
+
σ
2
θ
W
(
e
2
θ
t
)
e
−
θ
t
{\displaystyle r_{t}=\mu +{\sigma \over {\sqrt {2\theta }}}W(e^{2\theta t})e^{-\theta t}}
or conditionally (given
r
0
{\displaystyle r_{0}}
r
t
=
r
0
e
−
θ
t
+
μ
(
1
−
e
−
θ
t
)
+
σ
2
θ
W
(
e
2
θ
t
−
1
)
e
−
θ
t
.
{\displaystyle r_{t}=r_{0}e^{-\theta t}+\mu (1-e^{-\theta t})+{\sigma \over {\sqrt {2\theta }}}W(e^{2\theta t}-1)e^{-\theta t}.}
The Ornstein-Uhlenbeck process (an example of a Gaussian process that has a bounded variance) admits a stationary probability distribution , in contrast to the Wiener process .
f power spectrum.
Alternative representations
If B is a Brownian motion, then
U
t
=
exp
(
β
t
)
B
(
1
−
e
−
2
β
t
2
β
)
{\displaystyle U_{t}=\exp(\beta t)B\left({\frac {1-e^{-2\beta t}}{2\beta }}\right)}
defines an OU process and solves the equation
d
U
t
=
β
U
t
d
t
+
d
W
t
{\displaystyle dU_{t}=\beta U_{t}\,dt+dW_{t}}
where
W
{\displaystyle W}
See Also
The Vasicek model of interest rates is an example of an Ornstein-Uhlenbeck process.
![{\displaystyle \operatorname {cov} (r_{s},r_{t})=E[(r_{s}-E[r_{s}])(r_{t}-E[r_{t}])]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f73c2eb071f38823fc6754a4ff498c7e97729570)
![{\displaystyle =E[\int _{0}^{s}\sigma e^{\theta (u-s)}\,dW_{u}\int _{0}^{t}\sigma e^{\theta (v-t)}\,dW_{v}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f35a4db100fddd8c81f246ff167903133feb8e)
![{\displaystyle =\sigma ^{2}e^{-\theta (s+t)}E[\int _{0}^{s}e^{\theta u}\,dW_{u}\int _{0}^{t}e^{\theta v}\,dW_{v}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b11c7c31acbf43627451579b79fedabe69bd1369)
