Let $(X_{i})_{i\geq 1}$ be a sequence of i.i.d r.v with $P(X_{i} = 1) = p , P(X_{i} = -1) = 1-p,$ where $0 < p < 1$ and $p\neq 0.5$.
let $S_{n} = \sum\limits_{i=1}^n X_i $ and $M_n = (\frac{1-p}{p})^{S_n}$ .let A be an $F_m$ measurable event for some m. $A^{C}$ be the complement of A and let the constants $a,b\geq m$
Ive already shown that $M_n$ is a martingale w.r.t $(F_n)_{n\geq1}$ Now i need to Show that $E[M_{\tau}] = E[M_{m}]$ where $\tau = a 1_{A} + b 1_{A^{C}}$
My attempt : $E[M_{\tau}] = E[(\frac{1-p}{p})^{\sum\limits_{i=1}^\tau X_i}] = E[(\frac{1-p}{p})^{\sum\limits_{i=m+1}^\tau X_i +\sum\limits_{i=1}^mX_i}]= E[(\frac{1-p}{p})^{\sum\limits_{i=1}^mX_i}]E[(\frac{1-p}{p})^{\sum\limits_{i=m+1}^{\tau} X_i}] = E[M_m]$ if we can show that $E[(\frac{1-p}{p})^{\sum\limits_{i=m+1}^{\tau} X_i}] = 1$
$E[(\frac{1-p}{p})^{\sum\limits_{i=m+1}^{\tau} X_i}] = E[(\frac{1-p}{p})^{\sum\limits_{i=m+1}^a X_i1_{A} + \sum\limits_{i=m+1}^b X_i1_{A}}] $
I'm not sure how to proceed or if I'm even going about it correctly. Any help would be much appreciated.