# 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

$dx=a(r-x_{t})dt+\sigma dW_{t}$

## Implementation¶

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

1. 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.

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

3. 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.

4. 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.

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

$E(x_{0},t_{0},\Delta t)+S(x_{0},t_{0},\Delta t)*\Delta w$

where $$E$$ is the expectation and $$S$$ the standard deviation.