# Dimensionality in Monte Carlo Simulation

Given stochastic differential equation $$dx = \mu(x,t)dt+\sigma_1(x,t)dB_1+\sigma_2(x,t)dB_2 \tag1$$ where $dB_1$ and $B_2$ are orthogonal 1-dimentional Brownian motions. It is equivalent to $$dx = \mu(x,t)dt+\sigma(x,t)dB_3 \tag2$$ where $\sigma = \sqrt{\sigma_1^2+\sigma_2^2}.$

We are to solve these equations by Monte Carlo simulation. Equation (1) needs simulation of two Brownian motion whereas Equation (2) needs one.

Questions:

1) What is the comparison in error measure between the two versions?

2) Is there a substantial saving in running time and memory using Equation (2) over Equation (1)?