In this case, how do you know the average of the noise will be the desired one? In the box-muller is there any option to give the average amount? (I want the average of this noise at any period of time to be a certain given amount)
Box Muller will give you values of a normalized distribution (mean = 0, stdev=1), so you need to multiply with the desired standard deviation and add the desired offset to get a different mean. You simply need to scale it as required.