Im trying to simulate brownian motion and brownian motion with drift and i know they have properties like:
$ B(0) = 0 \ $ almost surely
$ B(t) \ \text{has distribution } N(0,t \sigma ^2) $
But what happens when the initial value of the BM is not 0? How does that affect the expected value and variance. Also im trying to simulate the BM as the limit of a random walk so i dont know how to handle the RW with a initial value different than 0