Correlated Wiener Processes

initial value

red process

0

green process

0

drift coefficient

red process

0

green process

0

diffusion coefficient

red process

0.3

green process

0.5

correlation coefficient

0

randomize

This Demonstration displays the paths of two correlated Wiener processes.

The concept of correlated stochastic processes is extremely important, particularly in areas of finance such as portfolio theory, but it can be somewhat counterintuitive, since highly correlated processes with very different diffusion and drift coefficients can look very different. You can gain an intuitive understanding of this concept by varying each of the parameters independently and observing the changes in the shape of the correlated paths.

Details

There are several ways to generate correlated processes. This Demonstration reveals the following simple fact: if

(

N

1

,

N

2

) are a pair of uncorrelated standard normal random variables then



ρ+1

N

1

2

+

1-ρ

N

2

,

ρ+1

N

1

2

-

1-ρ

N

2



are a pair of standard normal variables with correlation coefficient

ρ

.

External Links

Wiener Process (Wolfram MathWorld)

Permanent Citation

Andrzej Kozlowski

"Correlated Wiener Processes"
http://demonstrations.wolfram.com/CorrelatedWienerProcesses/
Wolfram Demonstrations Project
Published: March 7, 2011