Absolute value martingale of a simple symmetric random walk ($M_n=|S_n|-{\sum }_{k=0}^{n-1}{1}_{\{S_k=0\}}$)

Let a simple symmetric random walk start from a deterministic integer $S_0=i$, and put $L_n={\sum }_{k=0}^{n-1}{1}_{\{S_k=0\}}$. Then $M_n=|S_n|-L_n$ is a martingale with $M_0=|i|$. The visit count includes time $0$ if $i=0$.

For every integer $s$ and step $\xi \in \{-1,1\}$, the pathwise identityholds with $\mathrm{sgn}(0)=0$. Subtract the indicator function and sum over time to obtainThe sign function of the current state is a bounded predictable process for the next increment, so the sum is a martingale transform. Alternatively, averaging the two possible next absolute values shows directly that the conditional drift is $1_{\{S_k=0\}}$. Bounds $|S_n|\le |i|+n$ and $L_n\le n$ verify integrability at each finite time.

This is an absolute-value version of the discrete Tanaka formula. The correction records the occasions when the random walk is at the corner of the absolute-value function; away from zero that function is locally linear along a single step.