Past exam of the probability, measure and martingales course of the University of Oxford / 2021 / 3 / b / i / Solution

The filtration is generated by the successive independent random variables $X_k$. Each $S_n$ is adapted, and $|S_n|\le n$ makes it an integrable random variable. Since independence gives $\mathbb{E}[X_{n+1}\mid {\mathcal{F}}_n]=0$,Thus $(S_n)$ is a martingale.

Write the accumulated number of visits to zero asIt is adapted, and $0\le L_n\le n$. Thus $M_n=|S_n|-L_n$ is also adapted and an integrable random variable. For every integer $s$,Conditioning on $S_n$ and using independence of the next step therefore givesThe drift of the absolute value is cancelled exactly by the visit count, so $\mathbb{E}[M_{n+1}-M_n\mid {\mathcal{F}}_n]=0$. Hence both $(S_n)$ and $(M_n)$ are martingales. The latter is the absolute value martingale of a simple symmetric random walk, a discrete Tanaka formula in which visits to zero supply the correction at the corner of the absolute-value function.