multiple brownian motions construction

Multiple Brownian Motions Construction

Overview

Correlated Brownian Motions can be constructed from a set of uncorrelated Brownian Motions. Given a set of uncorrelated Brownian Motions, {% B_1,B_2 %}

{% dX = a_1 d B_1 + a_2 d B_2 %}

{% dy = a_3 d B_1 + a_4 d B_2 %}

{% (dX)(dY) = a_1 a_3 (dB_1)^2 + 2 a_1 a_4 (dB_1)(dB_2) + a_2 a_4 (dB_2)^2 %}

which given the rules

{% dB_1 dB_2 = 0 %}
{% dB^2 = dt %}

We then have

{% (dX)(dY) = (a_1 a_3 + a_2 a_4) dt %}

showing that

{% \rho = (a_1 a_3 + a_2 a_4) %}