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CoxIngersollRossProcess
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CoxIngersollRossProcess

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represents a Cox–Ingersoll–Ross process with long‐term mean μ, volatility σ, speed of adjustment θ, and initial condition x0.

Details

Examples

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Basic Examples  (3)Summary of the most common use cases

Simulate a Cox–Ingersoll–Ross process:

In[1]:=1
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-tzhca0
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-b4adct
Out[2]=2

Mean and variance functions:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-j0ussi
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-jrcqcv
Out[2]=2

Covariance function:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-5kfgm6
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-76krta
Out[2]=2

Scope  (14)Survey of the scope of standard use cases

Basic Uses  (9)

Simulate an ensemble of random paths for a Cox–Ingersoll–Ross process:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-bexgpm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-mn15g5
Out[2]=2

Simulate with arbitrary precision:

In[1]:=1
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-big92j
Out[1]=1

Compare paths for different values of the drift parameter:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-2vtftp
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-1hrt14
Out[2]=2

Compare paths for different values of the volatility parameter:

In[1]:=1
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-7tifxi
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-mo6pzv
Out[2]=2

Compare paths for different values of the speed adjustment parameter:

In[1]:=1
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-sbqrn8
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-lg9qdx
Out[2]=2

Simulate a Cox–Ingersoll–Ross process with different starting points:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-0pbe4h
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-37l3wb
In[3]:=3
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-sl7ej3
Out[3]=3

Process parameter estimation:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-b4q52
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-e93vf3
Out[2]=2

Correlation function:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-45h9gs
Out[1]=1

Absolute correlation function:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-r4jeh4
Out[1]=1

Process Slice Properties  (5)

First-order probability density function for the slice distribution:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-2st24h
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-k1o1wh
In[3]:=3
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-oegzkg
Out[3]=3
In[15]:=15
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-ej3ae1
Out[15]=15

Multivariate slice distributions:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-o8jlev
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-bkhq1j
Out[2]=2

Compute the expectation of an expression:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-edmw6r
Out[1]=1

Calculate the probability of an event:

In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-miucta
Out[2]=2

Skewness and kurtosis:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-xgr42o
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-bmnu2m
Out[2]=2

Moment of order r:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-84cmmy
Out[1]=1

Generating functions:

In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-kupkj
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-6jz3hy
Out[3]=3

CentralMoment and its generating function:

In[4]:=4
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-eawm2x
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-j6j83y
Out[5]=5

FactorialMoment and its generating function:

In[6]:=6
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-o10xs2
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-qux587
Out[7]=7

Cumulant and its generating function:

In[8]:=8
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-fmfg53
Out[8]=8
In[9]:=9
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-sddn3s
Out[9]=9

Properties & Relations  (3)Properties of the function, and connections to other functions

A Cox–ingersoll–Ross process is not weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-qkojiz
Out[1]=1

Conditional cumulative distribution function:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-ih0g5c
Out[1]=1

A Cox–ingersoll–Ross process is a special ItoProcess:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-b8dpyf
Out[1]=1

As well as StratonovichProcess:

In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-fx97pi
Out[2]=2

Neat Examples  (3)Surprising or curious use cases

Simulate a Cox–Ingersoll–Ross process in two dimensions:

In[1]:=1
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-2tpc38
In[2]:=2
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-glj5f0
Out[2]=2

Simulate a Cox–Ingersoll–Ross process in three dimensions:

In[1]:=1
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-k9tdgg
In[2]:=2
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-qim6n8
Out[2]=2

Simulate 500 paths from a Cox–Ingersoll–Ross process:

In[1]:=1
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-q7hoj9

Take a slice at 1 and visualize its distribution:

In[2]:=2
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-8s9gfo
In[3]:=3
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https://wolfram.com/xid/0is5kqbx5kczzusum14y6-g5fe1v

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

In[4]:=4
✖
https://wolfram.com/xid/0is5kqbx5kczzusum14y6-kwifgz
Out[4]=4
Wolfram Research (2012), CoxIngersollRossProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html.
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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.

✖
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.

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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

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

BibTeX

@misc{reference.wolfram_2025_coxingersollrossprocess, author="Wolfram Research", title="{CoxIngersollRossProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html}", note=[Accessed: 28-April-2025 ]}

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@misc{reference.wolfram_2025_coxingersollrossprocess, author="Wolfram Research", title="{CoxIngersollRossProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html}", note=[Accessed: 28-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_coxingersollrossprocess, organization={Wolfram Research}, title={CoxIngersollRossProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html}, note=[Accessed: 28-April-2025 ]}

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@online{reference.wolfram_2025_coxingersollrossprocess, organization={Wolfram Research}, title={CoxIngersollRossProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/CoxIngersollRossProcess.html}, note=[Accessed: 28-April-2025 ]}