# Stochastic Process¶

## Overview¶

Stochastic process uses the Random Number Generator(RNG). It uses a given point $$(t,x)$$ and a time step $$\Delta t$$ to calculate the expectation and variance. The class StochasticProcess1D is 1-dimensional stochastic process, and it cooperates with Cox-Ingersoll-Ross and Extended Cox-Ingersoll-Ross Models. The stochastic process can be described as

$dx_{t}=\mu(t,x_{t})dt+\sigma(t,x_{t})dW_{t}$

## Implementation¶

The implementation of StochasticProcess1D is comprised by a few methods. The implementation is introduced as follows:

1. init: Initialization function used to set up the 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. variance: The variance method returns the variance $$V(x_{t_{0}+\Delta t}|x_{t_{0}}=x_{0})$$ of the process during a time interval $$\Delta t$$ according to the given volatility.