Let $X_{i},i=1,2,...$ be a sequence of independent identically standard normally distributed random variables. Let $\{\mathcal{F}_{n},n\in\mathbb{N}\}$ be the natural filtration and $S_{n}=\sum^{n}_{i=1}X_{i}$ for $n\geq1$.
Compute $\mathbb{E}(e^{4X_{1}+2X_{2}}|X_{1}-X_{2})$.
My attempt:
$\mathbb{E}(e^{4X_{1}+2X_{2}}|X_{1}-X_{2})=\mathbb{E}(e^{4(X_{1}-X_{2})}e^{6X_{2}}|X_{1}-X_{2})=e^{4(X_{1}-X_{2})}\mathbb{E}(e^{6X_{2}}|X_{1}-X_{2})$ by taking out what is known, since the state $X_{1}-X_{2}$ is known. However, I would like to assume that $X_{2}$ is independent from $X_{1}-X_{2}$ to solve the computation, however this does not seem logical or I am not able to proof this independence. Is this the right way to do the computation, if yes, how do I proof the independence mentioned above?
If $X_1,X_2$ are independent normal random variables then they have a joint normal distribution and $Y_1=X_1+X_2$, $Y_2=X_1-X_2$ also have a joint normal distribution. You easily check that their covariance is $0$ in your case so they are independent. Now writing $X_1$ and $X_4$ in terms of $Y_1$ and $Y_2$ the required conditional expectation becomes $E(e^{3Y_1+Y_2}|Y_2)$ which is $e^{Y_2} Ee^{3Y_1}=e^{9} e^{X_1-X_2}$.