I am working on the following problem:
Let $Y_n$ be a sequence for which there exists constants $\alpha$ and $\beta$ with $$ E(Y_{n+1}\mid \mathcal{F}_n)=\alpha Y_n +\beta Y_{n-1} $$ for each $n$. Show that there exists a real number $a$ such that $X_n=aY_n+Y_{n-1}$ is a martingale.
Here the problem doesn't state it, but I am assuming that $(\mathcal{F}_n)_{n=1}^\infty$ is a filtration and $Y_n\in\mathcal{F}_n$.
This is my work so far. If I want $X_n$ to be a martingale then I need $E(X_{n+1}\mid\mathcal{F}_n)=X_n$. So we need $E(aY_{n+1}+Y_n\mid\mathcal{F}_n)=aY_n+Y_{n-1}$, but $$ E(aY_{n+1}+Y_n\mid\mathcal{F}_n)=aE(Y_{n+1}\mid\mathcal{F}_n)+Y_n=(a\alpha+1) Y_n + a\beta Y_{n-1} $$
So in order for $X_n$ to be a martingale we would need the constant $a$ to satisfy, $a=1/\beta,$ and $a=1/(1-\alpha).$
This is where I see an issue, unless $\alpha+\beta=1$ I don't see how such an $a$ can exist.
Is there something I am missing here?
For reference this problem is problem 9.12 from Probability, by Karr.