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I am trying to solve the next exercise about martingales.

Let $\{X_n,n\geq 1\}$ be a sequence of random variables with finite means and satisfying: $$E(X_{n+1}|X_0,X_1,...,X_n)=aX_n+bX_{n-1},\ n\geq 1$$ where $a>0,b<1$ and $a+b=1$. Find the value of $\alpha$ for which $S_n=\alpha X_n+X_{n-1}, \ n\geq 1$ is a martingale with respect to the natural filtration generated by the sequence $X_n$.

I have the next proposed solution:

If $S_n$ is a martingale I need that $E(S_n|\mathcal{F}_n)=S_{n-1}=\alpha X_{n-1}+X_{n-2}$.

Now I compute $E(S_n|\mathcal{F}_n)=\alpha X_{n-1}+\alpha(a+b)X_{n-2}+bX_{n-3}$

Using the previous expresion I don´t know how to finish it.

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  • $\begingroup$ It seems like you want your first expectation to be equal to something. Did you leave it out by mistake? $\endgroup$
    – Wintermute
    Commented Oct 11, 2017 at 13:29
  • $\begingroup$ Yes, I forgot, I have updated now. $\endgroup$ Commented Oct 11, 2017 at 13:42

1 Answer 1

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You are looking at $E(S_n \mid F_n)$, while you should be looking at $E(S_n \mid F_{n-1})$. Let's consider the following, observing that $X_n$ is $F_n$-measurable (as is $S_n$). $$ E(S_{n+1} \mid F_n) = E( \alpha X_{n+1} + X_n \mid F_n ) = \alpha E( X_{n+1} \mid F_n ) + X_n. $$ We now use the relation for $E(X_{n+1} \mid F_n)$ that you have to obtain $$ E(S_{n+1} \mid F_n) = \alpha \bigl( a X_n + b X_{n-1} \bigr) + X_n = (\alpha a + 1) X_n + \alpha b X_{n-1}. $$ We want this to be equal to $S_n = \alpha X_n + X_{n-1}$. I leave it to you to find what value(s) of $\alpha$ is(are) permissible. If you are still stuck, give me a shout :)

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