Local time of a semimartingale ($L_t^a(X)$)

For a continuous semimartingale $X$, symmetric local time $L_t^a(X)$ is an occupation density weighted by quadratic variation. Its normalization satisfies, for nonnegative Borel measurable functions $f$,It measures accumulated visits near a level on the scale of the process's fluctuations. In the symmetric Tanaka formula,use $\mathrm{sgn}(0)=0$. For Brownian motion, $d[X{]}_s=ds$, giving the Brownian local time occupation density. A deterministic smooth path instead has zero local time in this sense even if it spends a long ordinary time at one level.

At a reflecting boundary, finite-variation mass can be carried by the level itself. The right local time of a continuous semimartingale then differs from symmetric local time. Both satisfy the density identity, which determines values only for almost every spatial level; the chosen Tanaka formula fixes the convention at individual levels.

For Brownian motion with generator $\frac{1}{2}\Delta$, its local time is the occupation density normalized by ${\int }_0^{t}h(B_s)ds={\int }_{\mathbb{R}}h(a)L_t^{a}da$ for nonnegative Borel $h$. In the Tanaka formula it satisfies $|B_t-a|=|B_0-a|+{\int }_0^t\mathrm{sgn}(B_s-a)d{B}_s+L_t^a$. The symmetric-window approximation is $(2\epsilon {)}^{-1}{\int }_0^t{1}_{\{|B_s-a|<\epsilon \}}ds$. The factor two is essential. Occupation approximations for Brownian local time give a direct derivation at level zero from smooth approximations and maximal martingale estimates.

For Brownian motion started at zero, let $f_n$ equal $|x|$ outside $(-1/n,1/n)$ and $n{x}^2/2+1/(2n)$ inside. Its derivative is the clipped sign function $g_n(x)=\max \{-1,\min \{1,nx\}\}$. The Itô integral $M_t^{(n)}={\int }_0^t{g}_n(B_s)d{B}_s$ converges uniformly on bounded time intervals in $L^2$ to a continuous martingale $M$, by the Itô isometry and Doob L2 maximal inequality: the squared maximal error is at most $8\sqrt{2T/\pi }/n$. The Itô formula gives $\frac{n}{2}{\int }_0^t{1}_{\{|B_s|<1/n\}}ds=f_n(B_t)-f_n(0)-M_t^{(n)}$. Along $n=k^2$, the maximal errors are summable, so First Borel-Cantelli lemma gives uniform almost-sure convergence. The ratio-one interpolation for shrinking-window occupation integrals extends it to every integer $n$. The limit is $|B_t|-M_t=L_t^0$, the local time of a semimartingale in the Tanaka formula convention. The unnormalized factor $n$ gives twice this limit.

For any measurable real path $x$, put $A_n(t)=n{\int }_0^t{1}_{\{|x(s)|<1/n\}}ds$. If $n_k\le n\le n_{k+1}$, nested windows give $(n/n_{k+1})A_{n_{k+1}}(t)\le A_n(t)\le (n/n_k)A_{n_k}(t)$. Consequently, when $n_{k+1}/n_k\to 1$ and the subsequence $A_{n_k}$ converges uniformly on bounded time intervals, the full sequence does also, to the same limit. This deterministic interpolation turns summable-subsequence First Borel-Cantelli lemma estimates into almost-sure convergence without incorrectly claiming that convergence in $L^2$ alone implies full-sequence almost-sure convergence.

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.