A Short Note on Gaussian Distribution with Non-Constant Correlation

This article studies the terminal distribution of multi-variate Brownian motion where the correlations are not constant. In particular, with the assumption that the correlation function is driven by one factor, this article developed PDEs to quantify the moments of the conditional distribution of other factors. By using normal distribution and moment matching, we found a good approximation to the true Fokker Planck solution and the method provides a good analytic tractability and fast performance due to the low dimensions of PDEs to solve. This method can be applied to model correlation skew effect in quantitative finance, or other cases where a non-constant correlation is desired in modelling multi-variate distribution.