# QuantLib_OrnsteinUhlenbeckProcess man page

OrnsteinUhlenbeckProcess — Ornstein-Uhlenbeck process class.

## Synopsis

`#include <ql/processes/ornsteinuhlenbeckprocess.hpp>`

Inherits **StochasticProcess1D**.

### Public Member Functions

**OrnsteinUhlenbeckProcess** (**Real** speed, **Volatility** vol, **Real x0**=0.0, **Real** level=0.0)**Real x0** () const

returns the initial value of the state variable **Real speed** () const**Real volatility** () const**Real level** () const**Real variance** (**Time** t0, **Real x0**, **Time** dt) const

**StochasticProcess interface**

**Real drift** (**Time** t, **Real** x) const

returns the drift part of the equation, i.e. $ mu(t, x_t) $ **Real diffusion** (**Time** t, **Real** x) const

returns the diffusion part of the equation, i.e. $ sigma(t, x_t) $ **Real expectation** (**Time** t0, **Real x0**, **Time** dt) const**Real stdDeviation** (**Time** t0, **Real x0**, **Time** dt) const

### Additional Inherited Members

## Detailed Description

Ornstein-Uhlenbeck process class.

This class describes the Ornstein-Uhlenbeck process governed by [ dx = a (r - x_t) dt + sigma dW_t. ]

## Member Function Documentation

### Real expectation (Time t0, Real x0, Time dt) const [virtual]

returns the expectation $ E(x_{t_0 + Delta t} | x_{t_0} = x_0) $ of the process after a time interval $ Delta t $ according to the given discretization. This method can be overridden in derived classes which want to hard-code a particular discretization.

Reimplemented from **StochasticProcess1D**.

### Real stdDeviation (Time t0, Real x0, Time dt) const [virtual]

returns the standard deviation $ S(x_{t_0 + Delta t} | x_{t_0} = x_0) $ of the process after a time interval $ Delta t $ according to the given discretization. This method can be overridden in derived classes which want to hard-code a particular discretization.

Reimplemented from **StochasticProcess1D**.

### Real variance (Time t0, Real x0, Time dt) const [virtual]

returns the variance $ V(x_{t_0 + Delta t} | x_{t_0} = x_0) $ of the process after a time interval $ Delta t $ according to the given discretization. This method can be overridden in derived classes which want to hard-code a particular discretization.

Reimplemented from **StochasticProcess1D**.

## Author

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## Referenced By

OrnsteinUhlenbeckProcess(3) is an alias of QuantLib_OrnsteinUhlenbeckProcess(3).