ORNSTEIN-UHLENBECK PROCESS

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 ''r''_''t'' given by the following stochastic differential equation:
:$dr\_t\; =\; -\; heta\; (r\_t-mu),dt\; +\; sigma,\; dW\_t,,$
where θ, μ and σ are parameters and ''W''_''t'' denotes the Wiener process.
The Ornstein-Uhlenbeck process is the continuous-time analogue of the discrete-time AR(1) process.

Contents

Solution

Alternative representation I

Alternative representation II

References

See also

Generalisations

External links

Solution

This equation is solved by variation of parameters. Apply Itō's lemma to the function $f(r\_t,\; t)\; =\; r\_t\; e^\{\; heta\; t\}$ to get
:$df(r\_t,t)\; =\; heta\; r\_t\; e^\{\; heta\; t\},\; dt\; +\; e^\{\; heta\; t\},\; dr\_t,$
:$=\; e^\{\; heta\; t\}\; heta\; mu\; ,\; dt\; +\; sigma\; e^\{\; heta\; t\},\; dW\_t.\; ,$
Integrating from 0 to ''t'' we get
:$r\_t\; e^\{\; heta\; t\}\; =\; r\_0\; +\; int\_0^t\; e^\{\; heta\; s\}\; heta\; mu\; ,\; ds\; +\; int\_0^t\; sigma\; e^\{\; heta\; s\},\; dW\_s\; ,$
whereupon we see
:$r\_t\; =\; r\_0\; e^\{-\; heta\; t\}\; +\; mu(1-e^\{-\; heta\; t\})\; +\; int\_0^t\; sigma\; e^\{\; heta\; (s-t)\},\; dW\_s.\; ,$
Thus, the first moment is given by (assuming that $r\_0$ is a constant),
:$E(r\_t)=\; r\_0\; e^\{-\; heta\; t\}\; +\; mu(1-e^\{-\; heta\; t\}).$
Denote $s\; wedge\; t\; =\; min(s,t)$ we can use the Itō isometry to calculate the covariance function by
: $operatorname\{cov\}(r\_s,r\_t)=\; E[(r\_s\; -\; E[r\_s])(r\_t\; -\; E[r\_t])]$
:: $=\; E[int\_0^s\; sigma\; e^\{\; heta\; (u-s)\},\; dW\_u\; int\_0^t\; sigma\; e^\{\; heta\; (v-t)\},\; dW\_v\; ]$
:: $=\; sigma^2\; e^\{-\; heta\; (s+t)\}E[int\_0^s\; e^\{\; heta\; u\},\; dW\_u\; int\_0^t\; e^\{\; heta\; v\},\; dW\_v\; ]$
::

Alternative representation I

It is also possible (and often convenient) to represent $r\_t$ (unconditionally) as a scaled time-transformed Wiener process:
:$r\_t=mu+\{sigmaoversqrt\{2\; heta\}\}W(e^\{2\; heta\; t\})e^\{-\; heta\; t\}$
or conditionally (given $r\_0$) as
:$r\_t=r\_0\; e^\{-\; heta\; t\}\; +mu\; (1-e^\{-\; heta\; t\})+$
{sigmaoversqrt{2 heta}}W(e^{2 heta t}-1)e^{- heta 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.
The time integral of this process can be used to generate noise with a 1/''f'' power spectrum.

Alternative representation II

If ''B'' is a Brownian motion, then
:
ight)
defines an OU process and solves the equation
:
where $W$ is a Brownian motion. See Chamount and Yor for more.

References

★ ''G.E.Uhlenbeck and L.S.Ornstein'': "On the theory of Brownian Motion", Phys.Rev. 36:823-41, 1930

★ ''D.T.Gillespie'': "Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral", Phys.Rev.E 54:2084-91, 1996

See also

The Vasicek model of interest rates is an example of an Ornstein-Uhlenbeck process.

Generalisations

It is possible to extend the OU processes to processes where the background driving process is a Levy process. These processes are widely studied by Ole Barndorff-Nielsen and others.

External links

★ Simulating and Calibrating the Ornstein-Uhlenbeck process, sitmo.com