Tanaka's formula


In the stochastic calculus, Tanaka's formula for the Brownian motion states that
where Bt is the standard Brownian motion, sgn denotes the sign function
and Lt is its local time at 0 given by the L2-limit
One can also extend the formula to semimartingales.

Properties

Tanaka's formula is the explicit Doob-Meyer decomposition of the submartingale |Bt| into the martingale part, and a continuous increasing process. It can also be seen as the analogue of Itō's lemma for the absolute value function, with and ; see local time for a formal explanation of the Itō term.

Outline of proof

The function |x| is not C2 in x at x = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero by parabolas
and use Itō's formula, we can then take the limit as ε → 0, leading to Tanaka's formula.