# CoxIngersollRossProcess

CoxIngersollRossProcess[μ,σ,θ,x0]

represents a CoxIngersollRoss process with longterm mean μ, volatility σ, speed of adjustment θ, and initial condition x0.

# Details

• CoxIngersollRossProcess is also known as the CIR process.
• CoxIngersollRossProcess is a continuoustime and continuousstate random process.
• The state of the CoxIngersollRoss process satisfies an Ito differential equation , where follows a standard .
• CoxIngersollRossProcess allows x0 to be any positive real number, σ to be any nonzero real number, and θ and μ to be any nonzero real numbers of the same sign.
• CoxIngersollRossProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.

# Examples

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## Basic Examples(3)

Simulate a CoxIngersollRoss process:

Mean and variance functions:

Covariance function:

## Scope(14)

### Basic Uses(9)

Simulate an ensemble of random paths for a CoxIngersollRoss process:

Simulate with arbitrary precision:

Compare paths for different values of the drift parameter:

Compare paths for different values of the volatility parameter:

Compare paths for different values of the speed adjustment parameter:

Simulate a CoxIngersollRoss process with different starting points:

Process parameter estimation:

Correlation function:

Absolute correlation function:

### Process Slice Properties(5)

First-order probability density function for the slice distribution:

Multivariate slice distributions:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis:

Moment of order r:

Generating functions:

CentralMoment and its generating function:

FactorialMoment and its generating function:

Cumulant and its generating function:

## Properties & Relations(3)

A CoxingersollRoss process is not weakly stationary:

Conditional cumulative distribution function:

A CoxingersollRoss process is a special ItoProcess:

As well as StratonovichProcess:

## Neat Examples(3)

Simulate a CoxIngersollRoss process in two dimensions:

Simulate a CoxIngersollRoss process in three dimensions:

Simulate 500 paths from a CoxIngersollRoss process:

Take a slice at 1 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 1:

Wolfram Research (2012), CoxIngersollRossProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html.

#### Text

Wolfram Research (2012), CoxIngersollRossProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html.

#### CMS

Wolfram Language. 2012. "CoxIngersollRossProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html.

#### APA

Wolfram Language. (2012). CoxIngersollRossProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html

#### BibTeX

@misc{reference.wolfram_2024_coxingersollrossprocess, author="Wolfram Research", title="{CoxIngersollRossProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html}", note=[Accessed: 20-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_coxingersollrossprocess, organization={Wolfram Research}, title={CoxIngersollRossProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html}, note=[Accessed: 20-July-2024 ]}