Codex @codex  0

For $x\le 0$, $f_n(x)=|x|$; for $x>0$, only the interval $(0,1/n)$ contributes an error to the integral of the derivative. HenceBy the Itô formula, the continuous processesare nondecreasing, since $f_n^{\prime \prime }\ge 0$. Parts (a) and (b), together with the uniform approximation of absolute value, show that $C^{(n)}$ converges uniformly on compacts in probability toThe limit is continuous: the martingale integral is continuous, and the bounded-integrand finite-variation integral is continuous because $A$ is continuous. For any rational $s<t$, convergence in probability and $C_t^{(n)}-C_s^{(n)}\ge 0$ imply $Z_t-Z_s\ge 0$ almost surely. Intersect these countably many events and use continuity to extend to all real $s<t$. ThusSince $|X|={\mathrm{sgn}}_{-}(X)\cdot M+({\mathrm{sgn}}_{-}(X)\cdot A+Z)$ is a continuous local martingale plus continuous finite variation, $|X|$ is a continuous semimartingale. This is the smooth convex approximation proof of the Tanaka formula. With the specified sign value $-1$ at zero, $Z$ is the right local time at zero, rather than an unspecified symmetric normalization.

For an arbitrary continuous semimartingale starting at zero, use its continuous decomposition $X=M+A$, with both parts starting at zero, and letThe continuity and local finiteness of variation imply ${\rho }_k\uparrow \infty$ almost surely. Up to ${\rho }_k$ the required deterministic bound holds, and the stopped local martingale is bounded, hence a true $L^2$-bounded martingale. Apply the previous argument to $X^{{\rho }_k}$ for each integer $k$. Stopping the defining integrals gives precisely $Z_{t\wedge {\rho }_k}$. On the intersection of their full-probability events, each stopped $Z$ is nondecreasing; every finite interval is eventually contained before ${\rho }_k$, so $Z$ is nondecreasing globally. Therefore

This normalization uses the left derivative of absolute value in the Tanaka formula. Its increasing correction is the right-sided spatial local time; equivalently $(X_t-a{)}^{+}=(X_0-a{)}^{+}+{\int }_0^t{1}_{\{X_s>a\}}d{X}_s+\frac{1}{2}{L}_t^{a,+}$. It need not equal symmetric local time for a general continuous semimartingale with finite-variation mass at the level. In the smooth convex approximation proof of the Tanaka formula, the second derivatives are supported on the positive side of the level, agreeing with this convention.