# Ornstein-Uhlenbeck Process¶

## Overview¶

Ornstein-Uhlenbeck process is a stochastic process which uses the Random Number Generator (RNG) to generate locations for mesher. It uses a reference time point \(\Delta w\) and a time step \(\Delta t\) to calculate the drift and diffusion. The Ornstein-Uhlenbeck process is a simple stochastic processes, whose feature of interest is its mean-reverting drift term \(a(r-x)\) and its constant diffusion term \(\sigma\).

The Ornstein-Uhlenbeck process can be described by

## Implementation¶

The implementation of OrnsteinUhlenbeckProcess contains a few methods. The implementation can be introduced as follows:

init: The initialization process to set up arguments as below:

a)speed, the spreads on interest rates;

b)vola, the overall level of volatility;

c)x0, the initial value of level;

d)level, the width of fluctuation on interest rates.

expectation: The expectation method returns the expectation of the process at time \(E(x_{t_{0}+\Delta t}|x_{t_{0}}=x_{0})\).

stdDeviation: The stdDeviation method returns the standard deviation \(S(x_{t_{0}+\Delta t}|x_{t_{0}}=x_{0})\) of the process with a time period \(\Delta t\) according to the given discretization.

variance: The variance method returns the variance \(V(x_{t_{0}+\Delta t}|x_{t_{0}}=x_{0})\) of the process with a time period \(\Delta t\) according to the given discretization.

evolve: The evolve method returns the asset value after a time interval \(\Delta t\) according to the given discretization. It returns,

where \(E\) is the expectation and \(S\) the standard deviation.