Tanaka's formula


In the stochastic calculus, Tanaka's formula 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

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 using Itō's formula we can then take the limit as ε → 0, leading to Tanaka's formula.