Let $ X_1,X_2,... \sim i.i.d.$ $P(X_n=1)=p=1-P(X_n=-1)=1-q, n=1,2,...$, $S_n=\Sigma_{k=1}^{n}X_K$ and $F_n$ is filtration generated by $X_n$.
For fixed $\lambda>0$ we aim to find cosntant C that $Z_n = C^n\lambda^{S_n}$ is a martingale with respect to filtrafion $F_n$. We have to start with $$ E(Z_n |F_{n-1}) = E(C^n\lambda^{S_n} | F_{n-1}) = C^{n-1}E(\lambda^{X_1}\cdot ... \cdot \lambda^{X_{n-1}} | F_{n-1} )CE(\lambda^{X_n}|F_{n-1}) = Z_{n-1}CE(\lambda^{X_n}) $$ $$ = Z_{n-1}C(\lambda^1p+\lambda^{-1}q)$$ This has to equal $Z_{n-1}$ so $$C=(\lambda^1p+\lambda^{-1}q)^{-1}$$ Is this a great answer?
