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# Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck (OU) process satisfies the stochastic differential equation

\begin{eqnarray*}dX(t)&=&\kappa(\theta-X(t))dt+\sigma;dW(t),\\ X(0)&=&x_0 \end{eqnarray*} which has the solution \[ X(t)=\theta-e^{-\kappa t}(\theta-x_0)+\int_0^te^{-\kappa(t-s)}\sigma dW(s). \] The Ornstein-Uhlenbeck process is a continuous time counterpart to an AR(1)-process. More precisely if we look at the OU-process on equidistant grid $t_j=jh,j=0,1,2,\ldots$ we get the AR(1)-process \[X(t_{j+1})=\alpha(\theta-X(t_j))+e(t_{j+1}),\] where $\alpha=e^{-\kappa h}$ and where $e(t_{j+1})$ is a zero mean Gaussian random varible with variance $\sigma^2=(1-e^{-2\kappa h})/(2\kappa).$ Below you can see a simulation of the process where you can change the parameters.

The Ornstein-Uhlenbeck process is used as a model for short rates in the so called Vasicek model.

\begin{eqnarray*}dX(t)&=&\kappa(\theta-X(t))dt+\sigma;dW(t),\\ X(0)&=&x_0 \end{eqnarray*} which has the solution \[ X(t)=\theta-e^{-\kappa t}(\theta-x_0)+\int_0^te^{-\kappa(t-s)}\sigma dW(s). \] The Ornstein-Uhlenbeck process is a continuous time counterpart to an AR(1)-process. More precisely if we look at the OU-process on equidistant grid $t_j=jh,j=0,1,2,\ldots$ we get the AR(1)-process \[X(t_{j+1})=\alpha(\theta-X(t_j))+e(t_{j+1}),\] where $\alpha=e^{-\kappa h}$ and where $e(t_{j+1})$ is a zero mean Gaussian random varible with variance $\sigma^2=(1-e^{-2\kappa h})/(2\kappa).$ Below you can see a simulation of the process where you can change the parameters.

The Ornstein-Uhlenbeck process is used as a model for short rates in the so called Vasicek model.

Questions: Magnus Wiktorsson

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