Overview
Often times a model will require multliple Brownian Motions. The correlation between the Brownian motions is a function {% \rho(t) %} such that
{% \sum_{i=1}^N \Delta B_1(t_1) \times \Delta B_2(t_1) \rightarrow \int_0^T \rho(t)dt %}
Recalling the rule for a single Ito process
{% \sum_{i=1}^N \Delta X(t_i)^2 \rightarrow \int_0^T \sigma_x^2(t)dt %}
{% \sum_{i=1}^N \Delta X_1(t_i) \Delta X_2(t_i) \rightarrow \int_0^T \rho(t) \sigma_{x1}(t) \sigma_{x2}(t) dt %}
Differential Rule
A common way to manipulate stochastic process equations is the use of differntials. When dealing with multiple processes, the following differential is commonaly added to the list
{% dB_x dB_y = \rho dt %}
which is the analog of the 1-dimensional rule
{% (dB)^2 = dt %}