Let $(S_n)$ be a simple and symmetric random walk on $\Bbb Z$ and $(F_n)_{n≥0}$ its natural filtration. Find a sequence of real numbers $(\alpha_n)_n$ such that the process $(Y_n)_{n≥0}$ defined by $Y_n: = S^3_n + \alpha_n · S_n$ is a martingale.
I understand the idea is show that $E(Y_{n+1}|Y_n, ..., Y_0)=Y_n$, so replacing
$$ E(S^3_{n+1} + \alpha_{n+1} · S_{n+1}|Y_n, ..., Y_0)=E(S^3_{n+1} |Y_n, ..., Y_0)+E(\alpha_{n+1} · S_{n+1}|Y_n, ..., Y_0) $$ and I can remove the $\alpha_n$ because it is a sequence of real numbers, then i have $$ E(S^3_{n+1} + \alpha_{n+1} · S_{n+1}|Y_n, ..., Y_0)=E(S^3_{n+1} |Y_n, ..., Y_0)+ \alpha_{n+1}·E(S_{n+1}|Y_n, ..., Y_0) $$ but since $S_n$ is a symmetrical walk, this is a martingale. Finally $$ E(S^3_{n+1} + \alpha_{n+1} · S_{n+1}|Y_n, ..., Y_0)= S^3_n+\alpha_{n+1}·S_n $$ From all the above i conclude that $\alpha_{n+1}=\alpha_n$. Then it can be a constant sequence.
My proof is ok? I appreciate the help.
We must find a sequence $(\alpha_n)_{n \in\ \mathbb{N}}$ such that $$E[Y_{n+1}|\mathcal{A}_n]=Y_n \ \ \ \ \forall n \in \mathbb{N}$$ By taking the conditional expectations $$E[Y_{n+1}|\mathcal{A}_n]=E[S_{n+1}^3+S_{n+1}\alpha_{n+1}|\mathcal{A}_n]=E[S_{n+1}^3|\mathcal{A}_n]+\alpha_{n+1}S_n$$ $$E[Y_{n}|\mathcal{A}_n]=S_{n}^3+\alpha_{n}S_{n}$$ Ultimately we set $$E[Y_{n+1}-Y_n|\mathcal{A}_n]=E[S_{n+1}^3|\mathcal{A}_n]-S^3_n+(\alpha_{n+1}-\alpha_n)S_n=0$$ So we find recursively $$\alpha_{n+1}=\alpha_n-\frac{E[S_{n+1}^3|\mathcal{A}_n]-S^3_n}{S_n}$$