Given a probabilistic space $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n\geqslant 0},P)$. Let $X_n$ be a series of random variables respect to a given filtration, such that $E(X_i)=m \neq 0$, $E(X_i^2) \lt \inf$. Let's define $S_n = \sum_{i=1}^nX_i$. Find processes $A_n$, $B_n$ such that $S_n - A_n$ and $(S_n - A_n)^2 - B_n$ were martingales.
I've tried to calculate $S_n - A_n$, but I'm not sure if I did it correctly:
$$\mathbb{E}(S_{n+k}-A_{n+k}|\mathcal{F}_n)=\mathbb{E}(\sum_{i=1}^{n+k}X_i|\mathcal{F}_n)-\mathbb{E}(A_{n+k}|\mathcal{F}_n)=$$
$$\mathbb{E}(\sum_{i=1}^{n}X_i|\mathcal{F}_n)+\mathbb{E}(\sum_{i=n+1}^{n+k}X_i|\mathcal{F}_n)-\mathbb{E}(A_{n+k}|\mathcal{F}_n)=$$
$$\sum_{i=1}^nX_i+\mathbb{E}(\sum_{i=n+1}^{n+k}X_i)-\mathbb{E}(A_{n+k}|\mathcal{F}_n)=S_n+k\mu-\mathbb{E}(A_{n+k}|\mathcal{F}_n)$$
$$\Rightarrow A_n=k\mu-\mathbb{E}(A_{n+k}|\mathcal{F}_n)$$
$\Rightarrow A_n=-\sum_{i=1}^nY_i$, where $\mathbb{E}(Y_n)=-m$
I don't know how to calculate $(S_n - A_n)^2 - B_n$