I am given a probability space $(\Omega,F,P)$. Let $n=1,\dots,T$ $S_n=\exp\{\mu n+\sigma B_n\}$ with $B_n=\sum_{i=1}^n \epsilon_i$ with $\epsilon_i,i=1\dots,T$ independent standardnormal distributed random variables. Let $F_n=\sigma(B_1,\dots,B_n)$. Now $Q_u$ is defined by $\frac{dQ_u}{dP}=Y$ with $Y=\exp\{-uB_T-\frac{1}{2}u^2T\}$, $u \in \mathbb{R}$. After these assumptions I have to show, that $\epsilon_i$ are under $Q_u$ still independent and normal distributed and to compute $E_{Q_u}(\epsilon_i)$ and $Var_{Q_u}(\epsilon_i)$. Furthermore I should finally say for which $u$ ${S_n}$ is a martingale regarding $\{F_n\}$ and $Q_u$. However, I'd be glad if I first get to compute the expected value in order to conclude the last task.
First, I started with the definition and I think $\frac{dQ_u}{dP}=Y$ tells us that $Q_u(A)=\int_AYdP$, so for the independence I have to show that $$Q_u(\epsilon_i\cap\epsilon_j)=Q_u(\epsilon_i)Q_u(\epsilon_j), \ \ \ i\neq j$$ So $$Q_u(\epsilon_i\cap\epsilon_j)=\int_{\epsilon_i\cap\epsilon_j}YdP$$ Now I thought that I can use that they are independent and get $$\int_{\epsilon_i}YdP\int_{\epsilon_j}YdP=Q_u(\epsilon_i)Q_u(\epsilon_j).$$ Now in order to show that they are normal distributed I need to check the density function. However, I have no idea how to get it with the given information. Somehow I have never an idea where to start. Hope someone can help me. Thanks