In the stochastic calculus, Tanaka's formula for the Brownian motion states that
where B<sub>t</sub> is the standard Brownian motion, sgn denotes the sign function
and L<sub>t</sub> is its local time at 0 (the local time spent by B at 0 before time t) given by the L<sup>2</sup>-limit
One can also extend the formula to semimartingales.
Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale |B<sub>t</sub>| into the martingale part (the integral on the right-hand side, which is a Brownian motion), and a continuous increasing process (local time). It can also be seen as the analogue of Ità Â's lemma for the (nonsmooth) absolute value function , with and ; see local time for a formal explanation of the Ità  term.
The function |x| is not C<sup>2</sup> in x at x = 0, so we cannot apply Ità Â's formula directly. But if we approximate it near zero (i.e. in [−õ, õ]) by parabolas
and use Ità Â's formula, we can then take the limit as õ â 0, leading to Tanaka's formula.