Put $q=1-p$. Since
$$
\mathbb E\!\left[\lambda^{S_{n+1}}\mid\mathcal F_n\right]
=\lambda^{S_n}\left(p\lambda+q\lambda^{-1}\right),
$$
the <martingale> condition is
$$
p\lambda^2-\lambda+q=0
=(\lambda-1)(p\lambda-q).
$$
The root in $(0,1)$ is therefore
$$
\boxed{\lambda=\frac{1-p}{p}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-201/1/a/solution