# QuantLib_OrnsteinUhlenbeckProcess man page

OrnsteinUhlenbeckProcess — Ornstein-Uhlenbeck process class.

## Synopsis

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

Inherits **StochasticProcess1D**.

### Public Member Functions

OrnsteinUhlenbeckProcess(Realspeed,Volatilityvol,Real x0=0.0,Reallevel=0.0)Real x0() const

returns the initial value of the state variableReal speed() constReal volatility() constReal level() constReal variance(Timet0,Real x0,Timedt) const

**StochasticProcess interface**

Real drift(Timet,Realx) const

returns the drift part of the equation, i.e. $ mu(t, x_t) $Real diffusion(Timet,Realx) const

returns the diffusion part of the equation, i.e. $ sigma(t, x_t) $Real expectation(Timet0,Real x0,Timedt) constReal stdDeviation(Timet0,Real x0,Timedt) 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).