Tanaka's formula

Remarks

The function $y \mapsto \abs{y - x}$ is not twice differentiable at $x$, so Itô's lemma does not apply to it directly. Apply Itô's lemma to the convex functions $f_\varepsilon$ that agree with $\abs{y - x}$ outside $(x - \varepsilon, x + \varepsilon)$ and are quadratic with $f_\varepsilon'' = 1/\varepsilon$ inside. These functions are only $C^1$, but Itô's lemma still holds for them, by smoothing $f_\varepsilon$ further, because $f_\varepsilon''$ is bounded with two jumps. The second-order term is $\frac{1}{2\varepsilon} \int_0^t \mathbf{1}_{\set{\abs{B_s - x} < \varepsilon}} \, ds$, which tends to $L(t, x)$ as $\varepsilon \to 0$. The value chosen for $\sgn(0)$ does not matter, because Brownian motion spends a set of times of Lebesgue measure zero at $x$. Example: taking expectations with $x = 0$ gives $E[L(t, 0)] = E\abs{B_t} = \sqrt{2t/\pi}$.

Going deeper

For a student

The formula shows that $\abs{B}$ is a semimartingale: a martingale part plus an increasing part $L(\cdot, 0)$. By Lévy's characterization, the integral $W_t = \int_0^t \sgn(B_s) \, dB_s$ is a Brownian motion. Then $\abs{B} = W + L(\cdot, 0)$ and the Skorokhod reflection lemma give $L(t, 0) = \max_{s \le t} (-W_s)$. This leads to Lévy's identity in law between $\abs{B}$ and $M - B$, where $M$ is the running maximum. The same idea gives the Itô–Tanaka formula for differences of convex functions.