Variance of random walk Consider the Random Walk
$$X_t = X_{t-1}+\epsilon_t$$
with $\epsilon_t \sim N(\mu,\sigma^2)$ and $X_0=0$. We can write
$$X_t=X_0+\sum_{n=1}^t\epsilon_t.$$
Using the equation above we have
$$\mathbb{E}[X_t]=\mu t,$$
but I'm strugling to calculate the variance. I have
\begin{equation}
\begin{split}
Var[X_t] &=\mathbb{E}[X_t^2]-\mathbb{E}[X_t]^2 \\ 
& = \mathbb{E} \left[\left(X_0+\sum_{n=1}^t\epsilon_t\right)^2\right]-(\mu t)^2 \\ 
& = \mathbb{E} [X_0^2]-2X_0\mathbb{E}\left[\sum_{n=1}^t\epsilon_t\right]+\mathbb{E}\left[\left(\sum_{n=1}^t\epsilon_t\right)^2\right] - (\mu t)^2
\end{split}
\end{equation}
Well, now I think we can do
$$\mathbb{E}\left[\left(\sum_{n=1}^t\epsilon_t\right)^2\right]=\sigma^2 t, \mathbb{E}[X_0^2]=0$$
and
$$2X_0\mathbb{E}\left[\sum_{n=1}^t\epsilon_t\right]=2X_0\mu t=0$$
so we have
$$Var(X_t) = \sigma^2 t - (\mu t)^2 .$$
But I'm not sure that this is right. Any help would be appreciated!

Assuming the $\epsilon_t$ are iid, you can simply use
$$Var(X_t)=Var(\sum \epsilon_i)=\sum Var(\epsilon_i)$$
