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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.

For an integer $m$ and $d\in \{-1,0,1\}$, verify the single-step identityIf $m\ge 1$, the positive part is linear across the step and $f(m)=1$; if $m\le -1$, it remains zero and $f(m)=0$. At $m=0$, $d^{+}=(d+d^2)/2$, which is checked for the three permitted values of $d$.

Apply this with $m=S_{t-1}-K$ and $d=S_t-S_{t-1}$, and telescope over $t$. The result is the discrete Tanaka formulaThe first sum is a martingale transform; the second records the correction at the kink of the positive part. The algebraic identity itself needs only the integer values and permitted increments, not the martingale property.

Taking expectations in the single-step discrete Tanaka formula, the martingale transform term vanishes because $f$ is bounded and ${\mathcal{F}}_T$-measurable. HenceAt a state with $p_K=\mathbb{P}(S_T=K)>0$, the tower property of conditional expectation gives $\mathbb{E}[S_{T+1}\mid S_T=K]=K$, so the conditional second moment of the increment is ${\sigma }^2(T,K)$. Part (a) gives $p_K=C(T,K+1)-2C(T,K)+C(T,K-1)$. Combining these identities yields the discrete Dupire equationThe conditional variance needs a positive-probability conditioning state. The bound $|K-S_0|\le T$ does not guarantee $p_K>0$. If $p_K=0$, both the call-price curvature and the time increment above are zero; the equation can be extended by assigning an arbitrary finite value to ${\sigma }^2(T,K)$ at such a state, but its conditional variance is not determined there.

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.