Can someone explain the characteristics of the error term in a random walk? Why does the variance have to be constant? E(e) = 0 ?
A random walk represents a random variable. Random means erratic, something with no predictable forecast. For example, exchange rates are best defined or modeled as random walks. So why E(e)=0 ?, because this means that the best forecast of X(t) is x(t-1). Sounds weird, but the explanation in simple words is that due you have no way to forecast a random walk, its past value is the best predictor of the value today (at t). So on average, E(e) should be zero.
Thanks again!
Just to point out having an expected error term = 0 is different from having an error term with a constant variance.
They are two separate things 
The distribution of the error term in a random walk has both a mean of zero and a constant variance, and is uncorrelated with its value in the previous period.