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# Ornstein-Uhlenbeck process

# Definition

The *Ornstein-Uhlenbeck* process is a stochastic process
that satisfies the following stochastic differential equation:

$\displaystyle dX_{t}=\kappa(\theta-X_{t})\,dt+\sigma\,dW_{t}\,,$ | (1) |

where $W_{t}$ is a standard Brownian motion on $t\in[0,\infty)$.

# Mean-reverting property

If we ignore the random fluctuations in the process due to $dW_{t}$, then we see that $X_{t}$ has an overall drift towards a mean value $\theta$. The process $X_{t}$ reverts to this mean exponentially, at rate $\kappa$, with a magnitude in direct proportion to the distance between the current value of $X_{t}$ and $\theta$.

This can be seen by looking at the solution to the ordinary differential equation $dx_{t}=\kappa(\theta-x)dt$ which is

$\displaystyle\frac{\theta-x_{t}}{\theta-x_{0}}=e^{{-\kappa(t-t_{0})}}\,,\quad% \text{ or }x_{t}=\theta+(x_{0}-\theta)e^{{-\kappa(t-t_{0})}}\,.$ | (2) |

# Solution

The solution to the stochastic differential equation (1) defining the Ornstein-Uhlenbeck process is, for any $0\leq s\leq t$, is

$X_{t}=\theta+(X_{s}-\theta)e^{{-\kappa(t-s)}}+\sigma\int_{s}^{t}e^{{-\kappa(t-% u)}}\,dW_{u}\,.$ |

For any fixed $s$ and $t$, the random variable $X_{t}$, conditional upon $X_{s}$, is normally distributed with

$\text{mean}=\theta+(X_{s}-\theta)e^{{-\kappa(t-s)}}\,,\quad\text{variance}=% \frac{\sigma^{2}}{2\kappa}(1-e^{{-2\kappa(t-s)}})\,.$ |

Observe that the mean of $X_{t}$ is exactly the value derived heuristically in the solution (2) of the ODE.

The Ornstein-Uhlenbeck process is a time-homogeneous Itô diffusion.

# Applications

The Ornstein-Uhlenbeck process is widely used for modelling biological processes such as neuronal response, and in mathematical finance, the modelling of the dynamics of interest rates and volatilities of asset prices.

# References

- 1 Martin Jacobsen. “Laplace and the Origin of the Ornstein-Uhlenbeck Process”. Bernoulli, Vol. 2, No. 3. (Sept. 1996), pp. 271 – 286.
- 2
Bernt Øksendal.
*Stochastic Differential Equations, An Introduction with Applications*, 5th edition. Springer, 1998. - 3
Steven E. Shreve.
*Stochastic Calculus for Finance II: Continuous-Time Models*. Springer, 2004. - 4
Sebastian Jaimungal. Lecture notes for
*Pricing Theory*. University of Toronto. - 5
Dmitri Rubisov. Lecture notes for
*Risk Management*. University of Toronto.

## Mathematics Subject Classification

60H10*no label found*60-00

*no label found*

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